Differentiable Manifolds#

Given a non-discrete topological field \(K\) (in most applications, \(K = \RR\) or \(K = \CC\); see however [Ser1992] for \(K = \QQ_p\) and [Ber2008] for other fields), a differentiable manifold over \(K\) is a topological manifold \(M\) over \(K\) equipped with an atlas whose transitions maps are of class \(C^k\) (i.e. \(k\)-times continuously differentiable) for a fixed positive integer \(k\) (possibly \(k=\infty\)). \(M\) is then called a \(C^k\)-manifold over \(K\).

Note that

if the mention of \(K\) is omitted, then \(K=\RR\) is assumed;
if \(K=\CC\), any \(C^k\)-manifold with \(k\geq 1\) is actually a \(C^\infty\)-manifold (even an analytic manifold);
if \(K=\RR\), any \(C^k\)-manifold with \(k\geq 1\) admits a compatible \(C^\infty\)-structure (Whitney’s smoothing theorem).

Differentiable manifolds are implemented via the class DifferentiableManifold. Open subsets of differentiable manifolds are also implemented via DifferentiableManifold, since they are differentiable manifolds by themselves.

The user interface is provided by the generic function Manifold(), with the argument structure set to 'differentiable' and the argument diff_degree set to \(k\), or the argument structure set to 'smooth' (the default value).

Example 1: the 2-sphere as a differentiable manifold of dimension 2 over \(\RR\)

One starts by declaring \(S^2\) as a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'S^2')
sage: M
2-dimensional differentiable manifold S^2

Since the base topological field has not been specified in the argument list of Manifold, \(\RR\) is assumed:

sage: M.base_field()
Real Field with 53 bits of precision
sage: dim(M)
2

By default, the created object is a smooth manifold:

sage: M.diff_degree()
 +Infinity

Let us consider the complement of a point, the “North pole” say; this is an open subset of \(S^2\), which we call \(U\):

sage: U = M.open_subset('U'); U
Open subset U of the 2-dimensional differentiable manifold S^2

A standard chart on \(U\) is provided by the stereographic projection from the North pole to the equatorial plane:

sage: stereoN.<x,y> = U.chart(); stereoN
Chart (U, (x, y))

Thanks to the operator <x,y> on the left-hand side, the coordinates declared in a chart (here \(x\) and \(y\)), are accessible by their names; they are Sage’s symbolic variables:

sage: y
y
sage: type(y)
<class 'sage.symbolic.expression.Expression'>

The South pole is the point of coordinates \((x,y)=(0,0)\) in the above chart:

sage: S = U.point((0,0), chart=stereoN, name='S'); S
Point S on the 2-dimensional differentiable manifold S^2

Let us call \(V\) the open subset that is the complement of the South pole and let us introduce on it the chart induced by the stereographic projection from the South pole to the equatorial plane:

sage: V = M.open_subset('V'); V
Open subset V of the 2-dimensional differentiable manifold S^2
sage: stereoS.<u,v> = V.chart(); stereoS
Chart (V, (u, v))

The North pole is the point of coordinates \((u,v)=(0,0)\) in this chart:

sage: N = V.point((0,0), chart=stereoS, name='N'); N
Point N on the 2-dimensional differentiable manifold S^2

To fully construct the manifold, we declare that it is the union of \(U\) and \(V\):

sage: M.declare_union(U,V)

and we provide the transition map between the charts stereoN = \((U, (x, y))\) and stereoS = \((V, (u, v))\), denoting by \(W\) the intersection of \(U\) and \(V\) (\(W\) is the subset of \(U\) defined by \(x^2+y^2\not=0\), as well as the subset of \(V\) defined by \(u^2+v^2\not=0\)):

sage: stereoN_to_S = stereoN.transition_map(stereoS,
....:                [x/(x^2+y^2), y/(x^2+y^2)], intersection_name='W',
....:                restrictions1= x^2+y^2!=0, restrictions2= u^2+v^2!=0)
sage: stereoN_to_S
Change of coordinates from Chart (W, (x, y)) to Chart (W, (u, v))
sage: stereoN_to_S.display()
u = x/(x^2 + y^2)
v = y/(x^2 + y^2)

We give the name W to the Python variable representing \(W=U\cap V\):

sage: W = U.intersection(V)

The inverse of the transition map is computed by the method inverse():

sage: stereoN_to_S.inverse()
Change of coordinates from Chart (W, (u, v)) to Chart (W, (x, y))
sage: stereoN_to_S.inverse().display()
x = u/(u^2 + v^2)
y = v/(u^2 + v^2)

At this stage, we have four open subsets on \(S^2\):

sage: M.subset_family()
Set {S^2, U, V, W} of open subsets of the 2-dimensional differentiable manifold S^2

\(W\) is the open subset that is the complement of the two poles:

sage: N in W or S in W
False

The North pole lies in \(V\) and the South pole in \(U\):

sage: N in V, N in U
(True, False)
sage: S in U, S in V
(True, False)

The manifold’s (user) atlas contains four charts, two of them being restrictions of charts to a smaller domain:

sage: M.atlas()
[Chart (U, (x, y)), Chart (V, (u, v)), Chart (W, (x, y)), Chart (W, (u, v))]

Let us consider the point of coordinates (1,2) in the chart stereoN:

sage: p = M.point((1,2), chart=stereoN, name='p'); p
Point p on the 2-dimensional differentiable manifold S^2
sage: p.parent()
2-dimensional differentiable manifold S^2
sage: p in W
True

The coordinates of \(p\) in the chart stereoS are computed by letting the chart act on the point:

sage: stereoS(p)
(1/5, 2/5)

Given the definition of \(p\), we have of course:

sage: stereoN(p)
(1, 2)

Similarly:

sage: stereoS(N)
(0, 0)
sage: stereoN(S)
(0, 0)

A differentiable scalar field on the sphere:

sage: f = M.scalar_field({stereoN: atan(x^2+y^2), stereoS: pi/2-atan(u^2+v^2)},
....:                    name='f')
sage: f
Scalar field f on the 2-dimensional differentiable manifold S^2
sage: f.display()
f: S^2 → ℝ
on U: (x, y) ↦ arctan(x^2 + y^2)
on V: (u, v) ↦ 1/2*pi - arctan(u^2 + v^2)
sage: f(p)
arctan(5)
sage: f(N)
1/2*pi
sage: f(S)
0
sage: f.parent()
Algebra of differentiable scalar fields on the 2-dimensional differentiable
 manifold S^2
sage: f.parent().category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces

A differentiable manifold has a default vector frame, which, unless otherwise specified, is the coordinate frame associated with the first defined chart:

sage: M.default_frame()
Coordinate frame (U, (∂/∂x,∂/∂y))
sage: latex(M.default_frame())
\left(U, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y }\right)\right)
sage: M.default_frame() is stereoN.frame()
True

A vector field on the sphere:

sage: w = M.vector_field(name='w')
sage: w[stereoN.frame(), :] = [x, y]
sage: w.add_comp_by_continuation(stereoS.frame(), W, stereoS)
sage: w.display() # display in the default frame (stereoN.frame())
w = x ∂/∂x + y ∂/∂y
sage: w.display(stereoS.frame())
w = -u ∂/∂u - v ∂/∂v
sage: w.parent()
Module X(S^2) of vector fields on the 2-dimensional differentiable
 manifold S^2
sage: w.parent().category()
Category of modules over Algebra of differentiable scalar fields on the
 2-dimensional differentiable manifold S^2

Vector fields act on scalar fields:

sage: w(f)
Scalar field w(f) on the 2-dimensional differentiable manifold S^2
sage: w(f).display()
w(f): S^2 → ℝ
on U: (x, y) ↦ 2*(x^2 + y^2)/(x^4 + 2*x^2*y^2 + y^4 + 1)
on V: (u, v) ↦ 2*(u^2 + v^2)/(u^4 + 2*u^2*v^2 + v^4 + 1)
sage: w(f) == f.differential()(w)
True

The value of the vector field at point \(p\) is a vector tangent to the sphere:

sage: w.at(p)
Tangent vector w at Point p on the 2-dimensional differentiable manifold S^2
sage: w.at(p).display()
w = ∂/∂x + 2 ∂/∂y
sage: w.at(p).parent()
Tangent space at Point p on the 2-dimensional differentiable manifold S^2

A 1-form on the sphere:

sage: df = f.differential() ; df
1-form df on the 2-dimensional differentiable manifold S^2
sage: df.display()
df = 2*x/(x^4 + 2*x^2*y^2 + y^4 + 1) dx + 2*y/(x^4 + 2*x^2*y^2 + y^4 + 1) dy
sage: df.display(stereoS.frame())
df = -2*u/(u^4 + 2*u^2*v^2 + v^4 + 1) du - 2*v/(u^4 + 2*u^2*v^2 + v^4 + 1) dv
sage: df.parent()
Module Omega^1(S^2) of 1-forms on the 2-dimensional differentiable
 manifold S^2
sage: df.parent().category()
Category of modules over Algebra of differentiable scalar fields on the
 2-dimensional differentiable manifold S^2

The value of the 1-form at point \(p\) is a linear form on the tangent space at \(p\):

sage: df.at(p)
Linear form df on the Tangent space at Point p on the 2-dimensional
 differentiable manifold S^2
sage: df.at(p).display()
df = 1/13 dx + 2/13 dy
sage: df.at(p).parent()
Dual of the Tangent space at Point p on the 2-dimensional differentiable
 manifold S^2

Example 2: the Riemann sphere as a differentiable manifold of dimension 1 over \(\CC\)

We declare the Riemann sphere \(\CC^*\) as a 1-dimensional differentiable manifold over \(\CC\):

sage: M = Manifold(1, 'ℂ*', field='complex'); M
1-dimensional complex manifold ℂ*

We introduce a first open subset, which is actually \(\CC = \CC^*\setminus\{\infty\}\) if we interpret \(\CC^*\) as the Alexandroff one-point compactification of \(\CC\):

sage: U = M.open_subset('U')

A natural chart on \(U\) is then nothing but the identity map of \(\CC\), hence we denote the associated coordinate by \(z\):

sage: Z.<z> = U.chart()

The origin of the complex plane is the point of coordinate \(z=0\):

sage: O = U.point((0,), chart=Z, name='O'); O
Point O on the 1-dimensional complex manifold ℂ*

Another open subset of \(\CC^*\) is \(V = \CC^*\setminus\{O\}\):

sage: V = M.open_subset('V')

We define a chart on \(V\) such that the point at infinity is the point of coordinate 0 in this chart:

sage: W.<w> = V.chart(); W
Chart (V, (w,))
sage: inf = M.point((0,), chart=W, name='inf', latex_name=r'\infty')
sage: inf
Point inf on the 1-dimensional complex manifold ℂ*

To fully construct the Riemann sphere, we declare that it is the union of \(U\) and \(V\):

sage: M.declare_union(U,V)

and we provide the transition map between the two charts as \(w=1/z\) on on \(A = U\cap V\):

sage: Z_to_W = Z.transition_map(W, 1/z, intersection_name='A',
....:                           restrictions1= z!=0, restrictions2= w!=0)
sage: Z_to_W
Change of coordinates from Chart (A, (z,)) to Chart (A, (w,))
sage: Z_to_W.display()
w = 1/z
sage: Z_to_W.inverse()
Change of coordinates from Chart (A, (w,)) to Chart (A, (z,))
sage: Z_to_W.inverse().display()
z = 1/w

Let consider the complex number \(i\) as a point of the Riemann sphere:

sage: i = M((I,), chart=Z, name='i'); i
Point i on the 1-dimensional complex manifold ℂ*

Its coordinates with respect to the charts Z and W are:

sage: Z(i)
(I,)
sage: W(i)
(-I,)

and we have:

sage: i in U
True
sage: i in V
True

The following subsets and charts have been defined:

sage: M.subset_family()
Set {A, U, V, ℂ*} of open subsets of the 1-dimensional complex manifold ℂ*
sage: M.atlas()
[Chart (U, (z,)), Chart (V, (w,)), Chart (A, (z,)), Chart (A, (w,))]

A constant map \(\CC^* \rightarrow \CC\):

sage: f = M.constant_scalar_field(3+2*I, name='f'); f
Scalar field f on the 1-dimensional complex manifold ℂ*
sage: f.display()
f: ℂ* → ℂ
on U: z ↦ 2*I + 3
on V: w ↦ 2*I + 3
sage: f(O)
2*I + 3
sage: f(i)
2*I + 3
sage: f(inf)
2*I + 3
sage: f.parent()
Algebra of differentiable scalar fields on the 1-dimensional complex
 manifold ℂ*
sage: f.parent().category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces

A vector field on the Riemann sphere:

sage: v = M.vector_field(name='v')
sage: v[Z.frame(), 0] = z^2
sage: v.add_comp_by_continuation(W.frame(), U.intersection(V), W)
sage: v.display(Z.frame())
v = z^2 ∂/∂z
sage: v.display(W.frame())
v = -∂/∂w
sage: v.parent()
Module X(ℂ*) of vector fields on the 1-dimensional complex manifold ℂ*

The vector field \(v\) acting on the scalar field \(f\):

sage: v(f)
Scalar field zero on the 1-dimensional complex manifold ℂ*

Since \(f\) is constant, \(v(f)\) is vanishing:

sage: v(f).display()
zero: ℂ* → ℂ
on U: z ↦ 0
on V: w ↦ 0

The value of the vector field \(v\) at the point \(\infty\) is a vector tangent to the Riemann sphere:

sage: v.at(inf)
Tangent vector v at Point inf on the 1-dimensional complex manifold ℂ*
sage: v.at(inf).display()
v = -∂/∂w
sage: v.at(inf).parent()
Tangent space at Point inf on the 1-dimensional complex manifold ℂ*

AUTHORS:

Eric Gourgoulhon (2015): initial version
Travis Scrimshaw (2016): review tweaks
Michael Jung (2020): tensor bundles and orientability
Matthias Koeppe (2021): refactoring of subsets code

REFERENCES:

class sage.manifolds.differentiable.manifold.DifferentiableManifold(n, name, field, structure, base_manifold=None, diff_degree=+Infinity, latex_name=None, start_index=0, category=None, unique_tag=None)#

Bases: TopologicalManifold

Differentiable manifold over a topological field \(K\).

Note that

if the mention of \(K\) is omitted, then \(K=\RR\) is assumed;
if \(K=\CC\), any \(C^k\)-manifold with \(k\geq 1\) is actually a \(C^\infty\)-manifold (even an analytic manifold);
if \(K=\RR\), any \(C^k\)-manifold with \(k\geq 1\) admits a compatible \(C^\infty\)-structure (Whitney’s smoothing theorem).

INPUT:

n – positive integer; dimension of the manifold
name – string; name (symbol) given to the manifold
field – field \(K\) on which the manifold is defined; allowed values are
- 'real' or an object of type RealField (e.g., RR) for a manifold over \(\RR\)
- 'complex' or an object of type ComplexField (e.g., CC) for a manifold over \(\CC\)
- an object in the category of topological fields (see Fields and TopologicalSpaces) for other types of manifolds
structure – manifold structure (see DifferentialStructure or RealDifferentialStructure)
base_manifold – (default: None) if not None, must be a differentiable manifold; the created object is then an open subset of base_manifold
diff_degree – (default: infinity) degree \(k\) of differentiability
latex_name – (default: None) string; LaTeX symbol to denote the manifold; if none is provided, it is set to name
start_index – (default: 0) integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g. coordinates in a chart
category – (default: None) to specify the category; if None, Manifolds(field).Differentiable() (or Manifolds(field).Smooth() if diff_degree = infinity) is assumed (see the category Manifolds)
unique_tag – (default: None) tag used to force the construction of a new object when all the other arguments have been used previously (without unique_tag, the UniqueRepresentation behavior inherited from ManifoldSubset, via TopologicalManifold, would return the previously constructed object corresponding to these arguments).

EXAMPLES:

A 4-dimensional differentiable manifold (over \(\RR\)):

sage: M = Manifold(4, 'M', latex_name=r'\mathcal{M}'); M
4-dimensional differentiable manifold M
sage: type(M)
<class 'sage.manifolds.differentiable.manifold.DifferentiableManifold_with_category'>
sage: latex(M)
\mathcal{M}
sage: dim(M)
4

Since the base field has not been specified, \(\RR\) has been assumed:

sage: M.base_field()
Real Field with 53 bits of precision

Since the degree of differentiability has not been specified, the default value, \(C^\infty\), has been assumed:

sage: M.diff_degree()
+Infinity

The input parameter start_index defines the range of indices on the manifold:

sage: M = Manifold(4, 'M')
sage: list(M.irange())
[0, 1, 2, 3]
sage: M = Manifold(4, 'M', start_index=1)
sage: list(M.irange())
[1, 2, 3, 4]
sage: list(Manifold(4, 'M', start_index=-2).irange())
[-2, -1, 0, 1]

A complex manifold:

sage: N = Manifold(3, 'N', field='complex'); N
3-dimensional complex manifold N

A differentiable manifold over \(\QQ_5\), the field of 5-adic numbers:

sage: N = Manifold(2, 'N', field=Qp(5)); N
2-dimensional differentiable manifold N over the 5-adic Field with
 capped relative precision 20

A differentiable manifold is of course a topological manifold:

sage: isinstance(M, sage.manifolds.manifold.TopologicalManifold)
True
sage: isinstance(N, sage.manifolds.manifold.TopologicalManifold)
True

A differentiable manifold is a Sage parent object, in the category of differentiable (here smooth) manifolds over a given topological field (see Manifolds):

sage: isinstance(M, Parent)
True
sage: M.category()
Category of smooth manifolds over Real Field with 53 bits of precision
sage: from sage.categories.manifolds import Manifolds
sage: M.category() is Manifolds(RR).Smooth()
True
sage: M.category() is Manifolds(M.base_field()).Smooth()
True
sage: M in Manifolds(RR).Smooth()
True
sage: N in Manifolds(Qp(5)).Smooth()
True

The corresponding Sage elements are points:

sage: X.<t, x, y, z> = M.chart()
sage: p = M.an_element(); p
Point on the 4-dimensional differentiable manifold M
sage: p.parent()
4-dimensional differentiable manifold M
sage: M.is_parent_of(p)
True
sage: p in M
True

The manifold’s points are instances of class ManifoldPoint:

sage: isinstance(p, sage.manifolds.point.ManifoldPoint)
True

Since an open subset of a differentiable manifold \(M\) is itself a differentiable manifold, open subsets of \(M\) have all attributes of manifolds:

sage: U = M.open_subset('U', coord_def={X: t>0}); U
Open subset U of the 4-dimensional differentiable manifold M
sage: U.category()
Join of Category of subobjects of sets and Category of smooth manifolds
 over Real Field with 53 bits of precision
sage: U.base_field() == M.base_field()
True
sage: dim(U) == dim(M)
True

The manifold passes all the tests of the test suite relative to its category:

sage: TestSuite(M).run()

affine_connection(name, latex_name=None)#

Define an affine connection on the manifold.

See AffineConnection for a complete documentation.

INPUT:

name – name given to the affine connection
latex_name – (default: None) LaTeX symbol to denote the affine connection

OUTPUT:

the affine connection, as an instance of AffineConnection

EXAMPLES:

Affine connection on an open subset of a 3-dimensional smooth manifold:

sage: M = Manifold(3, 'M', start_index=1)
sage: A = M.open_subset('A', latex_name=r'\mathcal{A}')
sage: nab = A.affine_connection('nabla', r'\nabla') ; nab
Affine connection nabla on the Open subset A of the 3-dimensional
 differentiable manifold M

See also

AffineConnection for more examples.

automorphism_field(*comp, **kwargs)#

Define a field of automorphisms (invertible endomorphisms in each tangent space) on self.

Via the argument dest_map, it is possible to let the field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold and \(\Phi:\ M \rightarrow N\) a differentiable map, a field of automorphisms along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(1,1)} N\]

(\(T^{(1,1)} N\) being the tensor bundle of type \((1,1)\) over \(N\)) such that

\[\forall p \in M,\ t(p) \in \mathrm{GL}\left(T_{\Phi(p)} N \right),\]

where \(\mathrm{GL}\left(T_{\Phi(p)} N \right)\) is the general linear group of the tangent space \(T_{\Phi(p)} N\).

The standard case of a field of automorphisms on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

See also

AutomorphismField and AutomorphismFieldParal for a complete documentation.

INPUT:

comp – (optional) either the components of the field of automorphisms with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the field
latex_name – (default: None) LaTeX symbol to denote the field; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field of automorphisms on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a AutomorphismField (or if \(N\) is parallelizable, a AutomorphismFieldParal) representing the defined field of automorphisms

EXAMPLES:

A field of automorphisms on a 2-dimensional manifold:

sage: M = Manifold(2,'M')
sage: X.<x,y> = M.chart()
sage: a = M.automorphism_field([[1+x^2, 0], [0, 1+y^2]], name='A')
sage: a
Field of tangent-space automorphisms A on the 2-dimensional
 differentiable manifold M
sage: a.parent()
General linear group of the Free module X(M) of vector fields on
 the 2-dimensional differentiable manifold M
sage: a(X.frame()[0]).display()
A(∂/∂x) = (x^2 + 1) ∂/∂x
sage: a(X.frame()[1]).display()
A(∂/∂y) = (y^2 + 1) ∂/∂y

For more examples, see AutomorphismField and AutomorphismFieldParal.

automorphism_field_group(dest_map=None)#

Return the group of tangent-space automorphism fields defined on self, possibly with values in another manifold, as a module over the algebra of scalar fields defined on self.

If \(M\) is the current manifold and \(\Phi\) a differentiable map \(\Phi: M \rightarrow N\), where \(N\) is a differentiable manifold, this method called with dest_map being \(\Phi\) returns the general linear group \(\mathrm{GL}(\mathfrak{X}(M, \Phi))\) of the module \(\mathfrak{X}(M, \Phi)\) of vector fields along \(M\) with values in \(\Phi(M) \subset N\).

INPUT:

dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map, otherwise dest_map must be a DiffMap

OUTPUT:

a AutomorphismFieldParalGroup (if \(N\) is parallelizable) or a AutomorphismFieldGroup (if \(N\) is not parallelizable) representing \(\mathrm{GL}(\mathfrak{X}(U, \Phi))\)

EXAMPLES:

Group of tangent-space automorphism fields of a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: M.automorphism_field_group()
General linear group of the Module X(M) of vector fields on the
 2-dimensional differentiable manifold M
sage: M.automorphism_field_group().category()
Category of groups

See also

For more examples, see AutomorphismFieldParalGroup and AutomorphismFieldGroup.

change_of_frame(frame1, frame2)#

Return a change of vector frames defined on self.

INPUT:

frame1 – vector frame 1
frame2 – vector frame 2

OUTPUT:

a AutomorphismField representing, at each point, the vector space automorphism \(P\) that relates frame 1, \((e_i)\) say, to frame 2, \((n_i)\) say, according to \(n_i = P(e_i)\)

EXAMPLES:

Change of vector frames induced by a change of coordinates:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: c_uv.<u,v> = M.chart()
sage: c_xy.transition_map(c_uv, (x+y, x-y))
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))
sage: M.change_of_frame(c_xy.frame(), c_uv.frame())
Field of tangent-space automorphisms on the 2-dimensional
 differentiable manifold M
sage: M.change_of_frame(c_xy.frame(), c_uv.frame())[:]
[ 1/2  1/2]
[ 1/2 -1/2]
sage: M.change_of_frame(c_uv.frame(), c_xy.frame())
Field of tangent-space automorphisms on the 2-dimensional
 differentiable manifold M
sage: M.change_of_frame(c_uv.frame(), c_xy.frame())[:]
[ 1  1]
[ 1 -1]
sage: M.change_of_frame(c_uv.frame(), c_xy.frame()) == \
....:       M.change_of_frame(c_xy.frame(), c_uv.frame()).inverse()
True

In the present example, the manifold \(M\) is parallelizable, so that the module \(X(M)\) of vector fields on \(M\) is free. A change of frame on \(M\) is then identical to a change of basis in \(X(M)\):

sage: XM = M.vector_field_module() ; XM
Free module X(M) of vector fields on the 2-dimensional
 differentiable manifold M
sage: XM.print_bases()
Bases defined on the Free module X(M) of vector fields on the
 2-dimensional differentiable manifold M:
 - (M, (∂/∂x,∂/∂y)) (default basis)
 - (M, (∂/∂u,∂/∂v))
sage: XM.change_of_basis(c_xy.frame(), c_uv.frame())
Field of tangent-space automorphisms on the 2-dimensional
 differentiable manifold M
sage: M.change_of_frame(c_xy.frame(), c_uv.frame()) is \
....:  XM.change_of_basis(c_xy.frame(), c_uv.frame())
True

changes_of_frame()#

Return all the changes of vector frames defined on self.

OUTPUT:

dictionary of fields of tangent-space automorphisms representing the changes of frames, the keys being the pair of frames

EXAMPLES:

Let us consider a first vector frame on a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: e = X.frame(); e
Coordinate frame (M, (∂/∂x,∂/∂y))

At this stage, the dictionary of changes of frame is empty:

sage: M.changes_of_frame()
{}

We introduce a second frame on the manifold, relating it to frame e by a field of tangent space automorphisms:

sage: a = M.automorphism_field(name='a')
sage: a[:] = [[-y, x], [1, 2]]
sage: f = e.new_frame(a, 'f'); f
Vector frame (M, (f_0,f_1))

Then we have:

sage: M.changes_of_frame()  # random (dictionary output)
{(Coordinate frame (M, (∂/∂x,∂/∂y)),
  Vector frame (M, (f_0,f_1))): Field of tangent-space
   automorphisms on the 2-dimensional differentiable manifold M,
 (Vector frame (M, (f_0,f_1)),
  Coordinate frame (M, (∂/∂x,∂/∂y))): Field of tangent-space
   automorphisms on the 2-dimensional differentiable manifold M}

Some checks:

sage: M.changes_of_frame()[(e,f)] == a
True
sage: M.changes_of_frame()[(f,e)] == a^(-1)
True

coframes()#

Return the list of coframes defined on open subsets of self.

OUTPUT:

list of coframes defined on open subsets of self

EXAMPLES:

Coframes on subsets of \(\RR^2\):

sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: M.coframes()
[Coordinate coframe (R^2, (dx,dy))]
sage: e = M.vector_frame('e')
sage: M.coframes()
[Coordinate coframe (R^2, (dx,dy)), Coframe (R^2, (e^0,e^1))]
sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1}) # unit disk
sage: U.coframes()
[Coordinate coframe (U, (dx,dy))]
sage: e.restrict(U)
Vector frame (U, (e_0,e_1))
sage: U.coframes()
[Coordinate coframe (U, (dx,dy)), Coframe (U, (e^0,e^1))]
sage: M.coframes()
[Coordinate coframe (R^2, (dx,dy)),
 Coframe (R^2, (e^0,e^1)),
 Coordinate coframe (U, (dx,dy)),
 Coframe (U, (e^0,e^1))]

cotangent_bundle(dest_map=None)#

Return the cotangent bundle possibly along a destination map with base space self.

See also

TensorBundle for complete documentation.

INPUT:

dest_map – (default: None) destination map \(\Phi:\ M \rightarrow N\) (type: DiffMap) from which the cotangent bundle is pulled back; if None, it is assumed that \(N=M\) and \(\Phi\) is the identity map of \(M\) (case of the standard tangent bundle over \(M\))

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: cTM = M.cotangent_bundle(); cTM
Cotangent bundle T*M over the 2-dimensional differentiable
 manifold M

curve(coord_expression, param, chart=None, name=None, latex_name=None)#

Define a differentiable curve in the manifold.

See also

DifferentiableCurve for details.

INPUT:

coord_expression – either
- (i) a dictionary whose keys are charts on the manifold and values the coordinate expressions (as lists or tuples) of the curve in the given chart
- (ii) a single coordinate expression in a given chart on the manifold, the latter being provided by the argument chart
in both cases, if the dimension of the manifold is 1, a single coordinate expression can be passed instead of a tuple with a single element
param – a tuple of the type (t, t_min, t_max), where
- t is the curve parameter used in coord_expression;
- t_min is its minimal value;
- t_max its maximal value;
if t_min=-Infinity and t_max=+Infinity, they can be omitted and t can be passed for param instead of the tuple (t, t_min, t_max)
chart – (default: None) chart on the manifold used for case (ii) above; if None the default chart of the manifold is assumed
name – (default: None) string; symbol given to the curve
latex_name – (default: None) string; LaTeX symbol to denote the curve; if none is provided, name will be used

OUTPUT:

DifferentiableCurve

EXAMPLES:

The lemniscate of Gerono in the 2-dimensional Euclidean plane:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = manifolds.RealLine()
sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c') ; c
Curve c in the 2-dimensional differentiable manifold M

The same definition with the coordinate expression passed as a dictionary:

sage: c = M.curve({X: [sin(t), sin(2*t)/2]}, (t, 0, 2*pi), name='c') ; c
Curve c in the 2-dimensional differentiable manifold M

An example of definition with t_min and t_max omitted: a helix in \(\RR^3\):

sage: R3 = Manifold(3, 'R^3')
sage: X.<x,y,z> = R3.chart()
sage: c = R3.curve([cos(t), sin(t), t], t, name='c') ; c
Curve c in the 3-dimensional differentiable manifold R^3
sage: c.domain() # check that t is unbounded
Real number line ℝ

See also

DifferentiableCurve for more examples, including plots.

de_rham_complex(dest_map=None)#

Return the set of mixed forms defined on self, possibly with values in another manifold, as a graded algebra.

See also

MixedFormAlgebra for complete documentation.

INPUT:

dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of mixed forms on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a MixedFormAlgebra representing the graded algebra \(\Omega^*(M,\Phi)\) of mixed forms on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Graded algebra of mixed forms on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: M.mixed_form_algebra()
Graded algebra Omega^*(M) of mixed differential forms on the
 2-dimensional differentiable manifold M
sage: M.mixed_form_algebra().category()
Join of Category of graded algebras over Symbolic Ring and Category of chain complexes over Symbolic Ring
sage: M.mixed_form_algebra().base_ring()
Symbolic Ring

The outcome is cached:

sage: M.mixed_form_algebra() is M.mixed_form_algebra()
True

default_frame()#

Return the default vector frame defined on self.

By vector frame, it is meant a field on the manifold that provides, at each point \(p\), a vector basis of the tangent space at \(p\).

Unless changed via set_default_frame(), the default frame is the first one defined on the manifold, usually implicitly as the coordinate basis associated with the first chart defined on the manifold.

OUTPUT:

a VectorFrame representing the default vector frame

EXAMPLES:

The default vector frame is often the coordinate frame associated with the first chart defined on the manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: M.default_frame()
Coordinate frame (M, (∂/∂x,∂/∂y))

degenerate_metric(name, latex_name=None, dest_map=None)#

Define a degenerate (or null or lightlike) metric on the manifold.

A degenerate metric is a field of degenerate symmetric bilinear forms acting in the tangent spaces.

See DegenerateMetric for a complete documentation.

INPUT:

name – name given to the metric
latex_name – (default: None) LaTeX symbol to denote the metric; if None, it is formed from name
dest_map – (default: None) instance of class DiffMap representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; if None, the identity map is assumed (case of a metric tensor field on \(U\))

OUTPUT:

instance of DegenerateMetric representing the defined degenerate metric.

EXAMPLES:

Lightlike cone:

sage: M = Manifold(3, 'M'); X.<x,y,z> = M.chart()
sage: g = M.degenerate_metric('g'); g
degenerate metric g on the 3-dimensional differentiable manifold M
sage: det(g)
Scalar field zero on the 3-dimensional differentiable manifold M
sage: g.parent()
Free module T^(0,2)(M) of type-(0,2) tensors fields on the
3-dimensional differentiable manifold M
sage: g[0,0], g[0,1], g[0,2] = (y^2 + z^2)/(x^2 + y^2 + z^2), \
....: - x*y/(x^2 + y^2 + z^2), - x*z/(x^2 + y^2 + z^2)
sage: g[1,1], g[1,2], g[2,2] = (x^2 + z^2)/(x^2 + y^2 + z^2), \
....: - y*z/(x^2 + y^2 + z^2), (x^2 + y^2)/(x^2 + y^2 + z^2)
sage: g.disp()
g = (y^2 + z^2)/(x^2 + y^2 + z^2) dx⊗dx - x*y/(x^2 + y^2 + z^2) dx⊗dy
- x*z/(x^2 + y^2 + z^2) dx⊗dz - x*y/(x^2 + y^2 + z^2) dy⊗dx
+ (x^2 + z^2)/(x^2 + y^2 + z^2) dy⊗dy - y*z/(x^2 + y^2 + z^2) dy⊗dz
- x*z/(x^2 + y^2 + z^2) dz⊗dx - y*z/(x^2 + y^2 + z^2) dz⊗dy
+ (x^2 + y^2)/(x^2 + y^2 + z^2) dz⊗dz

See also

DegenerateMetric for more examples.

diff_degree()#

Return the manifold’s degree of differentiability.

The degree of differentiability is the integer \(k\) (possibly \(k=\infty\)) such that the manifold is a \(C^k\)-manifold over its base field.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: M.diff_degree()
+Infinity
sage: M = Manifold(2, 'M', structure='differentiable', diff_degree=3)
sage: M.diff_degree()
3

diff_form(*args, **kwargs)#

Define a differential form on self.

Via the argument dest_map, it is possible to let the differential form take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold, \(\Phi:\ M \rightarrow N\) a differentiable map and \(p\) a non-negative integer, a differential form of degree \(p\) (or \(p\)-form) along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(0,p)}N\]

(\(T^{(0,p)} N\) being the tensor bundle of type \((0,p)\) over \(N\)) such that

\[\forall x \in M,\quad t(x) \in \Lambda^p(T^*_{\Phi(x)} N),\]

where \(\Lambda^p(T^*_{\Phi(x)} N)\) is the \(p\)-th exterior power of the dual of the tangent space \(T_{\Phi(x)} N\).

The standard case of a differential form on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

For \(p = 1\), one can use the method one_form() instead.

See also

DiffForm and DiffFormParal for a complete documentation.

INPUT:

degree – the degree \(p\) of the differential form (i.e. its tensor rank)
comp – (optional) either the components of the differential form with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the differential form
latex_name – (default: None) LaTeX symbol to denote the differential form; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a differential form on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

the \(p\)-form as a DiffForm (or if \(N\) is parallelizable, a DiffFormParal)

EXAMPLES:

A 2-form on a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: f = M.diff_form(2, name='F'); f
2-form F on the 3-dimensional differentiable manifold M
sage: f[0,1], f[1,2] = x+y, x*z
sage: f.display()
F = (x + y) dx∧dy + x*z dy∧dz

For more examples, see DiffForm and DiffFormParal.

diff_form_module(degree, dest_map=None)#

Return the set of differential forms of a given degree defined on self, possibly with values in another manifold, as a module over the algebra of scalar fields defined on self.

See also

DiffFormModule for complete documentation.

INPUT:

degree – positive integer; the degree \(p\) of the differential forms
dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of differential forms on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a DiffFormModule (or if \(N\) is parallelizable, a DiffFormFreeModule) representing the module \(\Omega^p(M,\Phi)\) of \(p\)-forms on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Module of 2-forms on a 3-dimensional parallelizable manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: M.diff_form_module(2)
Free module Omega^2(M) of 2-forms on the 3-dimensional
 differentiable manifold M
sage: M.diff_form_module(2).category()
Category of finite dimensional modules over Algebra of
 differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: M.diff_form_module(2).base_ring()
Algebra of differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: M.diff_form_module(2).rank()
3

The outcome is cached:

sage: M.diff_form_module(2) is M.diff_form_module(2)
True

diff_map(codomain, coord_functions=None, chart1=None, chart2=None, name=None, latex_name=None)#

Define a differentiable map between the current differentiable manifold and a differentiable manifold over the same topological field.

See DiffMap for a complete documentation.

INPUT:

codomain – the map codomain (a differentiable manifold over the same topological field as the current differentiable manifold)
coord_functions – (default: None) if not None, must be either
- (i) a dictionary of the coordinate expressions (as lists (or tuples) of the coordinates of the image expressed in terms of the coordinates of the considered point) with the pairs of charts (chart1, chart2) as keys (chart1 being a chart on the current manifold and chart2 a chart on codomain)
- (ii) a single coordinate expression in a given pair of charts, the latter being provided by the arguments chart1 and chart2
In both cases, if the dimension of the arrival manifold is 1, a single coordinate expression can be passed instead of a tuple with a single element
chart1 – (default: None; used only in case (ii) above) chart on the current manifold defining the start coordinates involved in coord_functions for case (ii); if none is provided, the coordinates are assumed to refer to the manifold’s default chart
chart2 – (default: None; used only in case (ii) above) chart on codomain defining the arrival coordinates involved in coord_functions for case (ii); if none is provided, the coordinates are assumed to refer to the default chart of codomain
name – (default: None) name given to the differentiable map
latex_name – (default: None) LaTeX symbol to denote the differentiable map; if none is provided, the LaTeX symbol is set to name

OUTPUT:

the differentiable map, as an instance of DiffMap

EXAMPLES:

A differentiable map between an open subset of \(S^2\) covered by regular spherical coordinates and \(\RR^3\):

sage: M = Manifold(2, 'S^2')
sage: U = M.open_subset('U')
sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: N = Manifold(3, 'R^3', r'\RR^3')
sage: c_cart.<x,y,z> = N.chart()  # Cartesian coord. on R^3
sage: Phi = U.diff_map(N, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)),
....:                  name='Phi', latex_name=r'\Phi')
sage: Phi
Differentiable map Phi from the Open subset U of the 2-dimensional
 differentiable manifold S^2 to the 3-dimensional differentiable
 manifold R^3

The same definition, but with a dictionary with pairs of charts as keys (case (i) above):

sage: Phi1 = U.diff_map(N,
....:        {(c_spher, c_cart): (sin(th)*cos(ph), sin(th)*sin(ph),
....:         cos(th))}, name='Phi', latex_name=r'\Phi')
sage: Phi1 == Phi
True

The differentiable map acting on a point:

sage: p = U.point((pi/2, pi)) ; p
Point on the 2-dimensional differentiable manifold S^2
sage: Phi(p)
Point on the 3-dimensional differentiable manifold R^3
sage: Phi(p).coord(c_cart)
(-1, 0, 0)
sage: Phi1(p) == Phi(p)
True

See the documentation of class DiffMap for more examples.

diffeomorphism(codomain=None, coord_functions=None, chart1=None, chart2=None, name=None, latex_name=None)#

Define a diffeomorphism between the current manifold and another one.

See DiffMap for a complete documentation.

INPUT:

codomain – (default: None) codomain of the diffeomorphism (the arrival manifold or some subset of it). If None, the current manifold is taken.
coord_functions – (default: None) if not None, must be either
- (i) a dictionary of the coordinate expressions (as lists (or tuples) of the coordinates of the image expressed in terms of the coordinates of the considered point) with the pairs of charts (chart1, chart2) as keys (chart1 being a chart on the current manifold and chart2 a chart on codomain)
- (ii) a single coordinate expression in a given pair of charts, the latter being provided by the arguments chart1 and chart2
In both cases, if the dimension of the arrival manifold is 1, a single coordinate expression can be passed instead of a tuple with a single element
chart1 – (default: None; used only in case (ii) above) chart on the current manifold defining the start coordinates involved in coord_functions for case (ii); if none is provided, the coordinates are assumed to refer to the manifold’s default chart
chart2 – (default: None; used only in case (ii) above) chart on codomain defining the arrival coordinates involved in coord_functions for case (ii); if none is provided, the coordinates are assumed to refer to the default chart of codomain
name – (default: None) name given to the diffeomorphism
latex_name – (default: None) LaTeX symbol to denote the diffeomorphism; if none is provided, the LaTeX symbol is set to name

OUTPUT:

the diffeomorphism, as an instance of DiffMap

EXAMPLES:

Diffeomorphism between the open unit disk in \(\RR^2\) and \(\RR^2\):

sage: M = Manifold(2, 'M')  # the open unit disk
sage: forget()  # for doctests only
sage: c_xy.<x,y> = M.chart('x:(-1,1) y:(-1,1)', coord_restrictions=lambda x,y: x^2+y^2<1)
....:    # Cartesian coord on M
sage: N = Manifold(2, 'N')  # R^2
sage: c_XY.<X,Y> = N.chart()  # canonical coordinates on R^2
sage: Phi = M.diffeomorphism(N, [x/sqrt(1-x^2-y^2), y/sqrt(1-x^2-y^2)],
....:                        name='Phi', latex_name=r'\Phi')
sage: Phi
Diffeomorphism Phi from the 2-dimensional differentiable manifold M
 to the 2-dimensional differentiable manifold N
sage: Phi.display()
Phi: M → N
   (x, y) ↦ (X, Y) = (x/sqrt(-x^2 - y^2 + 1), y/sqrt(-x^2 - y^2 + 1))

The inverse diffeomorphism:

sage: Phi^(-1)
Diffeomorphism Phi^(-1) from the 2-dimensional differentiable
 manifold N to the 2-dimensional differentiable manifold M
sage: (Phi^(-1)).display()
Phi^(-1): N → M
   (X, Y) ↦ (x, y) = (X/sqrt(X^2 + Y^2 + 1), Y/sqrt(X^2 + Y^2 + 1))

See the documentation of class DiffMap for more examples.

frames()#

Return the list of vector frames defined on open subsets of self.

OUTPUT:

list of vector frames defined on open subsets of self

EXAMPLES:

Vector frames on subsets of \(\RR^2\):

sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: M.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y))]
sage: e = M.vector_frame('e')
sage: M.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y)),
 Vector frame (R^2, (e_0,e_1))]
sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1}) # unit disk
sage: U.frames()
[Coordinate frame (U, (∂/∂x,∂/∂y))]
sage: M.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y)),
 Vector frame (R^2, (e_0,e_1)),
 Coordinate frame (U, (∂/∂x,∂/∂y))]

integrated_autoparallel_curve(affine_connection, curve_param, initial_tangent_vector, chart=None, name=None, latex_name=None, verbose=False, across_charts=False)#

Construct an autoparallel curve on the manifold with respect to a given affine connection.

See also

IntegratedAutoparallelCurve for details.

INPUT:

affine_connection – AffineConnection; affine connection with respect to which the curve is autoparallel
curve_param – a tuple of the type (t, t_min, t_max), where
- t is the symbolic variable to be used as the parameter of the curve (the equations defining an instance of IntegratedAutoparallelCurve are such that t will actually be an affine parameter of the curve);
- t_min is its minimal (finite) value;
- t_max its maximal (finite) value.
initial_tangent_vector – TangentVector; initial tangent vector of the curve
chart – (default: None) chart on the manifold in which the equations are given ; if None the default chart of the manifold is assumed
name – (default: None) string; symbol given to the curve
latex_name – (default: None) string; LaTeX symbol to denote the curve; if none is provided, name will be used

OUTPUT:

IntegratedAutoparallelCurve

EXAMPLES:

Autoparallel curves associated with the Mercator projection of the 2-sphere \(\mathbb{S}^{2}\):

sage: S2 = Manifold(2, 'S^2', start_index=1)
sage: polar.<th,ph> = S2.chart('th ph')
sage: epolar = polar.frame()
sage: ch_basis = S2.automorphism_field()
sage: ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
sage: epolar_ON=S2.default_frame().new_frame(ch_basis,'epolar_ON')

Set the affine connection associated with Mercator projection; it is metric compatible but it has non-vanishing torsion:

sage: nab = S2.affine_connection('nab')
sage: nab.set_coef(epolar_ON)[:]
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
sage: g = S2.metric('g')
sage: g[1,1], g[2,2] = 1, (sin(th))^2
sage: nab(g)[:]
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
sage: nab.torsion()[:]
[[[0, 0], [0, 0]], [[0, cos(th)/sin(th)], [-cos(th)/sin(th), 0]]]

Declare an integrated autoparallel curve with respect to this connection:

sage: p = S2.point((pi/4, 0), name='p')
sage: Tp = S2.tangent_space(p)
sage: v = Tp((1,1), basis=epolar_ON.at(p))
sage: t = var('t')
sage: c = S2.integrated_autoparallel_curve(nab, (t, 0, 2.3),
....:                              v, chart=polar, name='c')
sage: sys = c.system(verbose=True)
Autoparallel curve c in the 2-dimensional differentiable
 manifold S^2 equipped with Affine connection nab on the
 2-dimensional differentiable manifold S^2, and integrated
 over the Real interval (0, 2.30000000000000) as a solution to the
 following equations, written with respect to
 Chart (S^2, (th, ph)):

Initial point: Point p on the 2-dimensional differentiable
 manifold S^2 with coordinates [1/4*pi, 0] with respect to
 Chart (S^2, (th, ph))
Initial tangent vector: Tangent vector at Point p on the
 2-dimensional differentiable manifold S^2 with
 components [1, sqrt(2)] with respect to
 Chart (S^2, (th, ph))

d(th)/dt = Dth
d(ph)/dt = Dph
d(Dth)/dt = 0
d(Dph)/dt = -Dph*Dth*cos(th)/sin(th)

sage: sol = c.solve()
sage: interp = c.interpolate()
sage: p = c(1.3, verbose=True)
Evaluating point coordinates from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: p
Point on the 2-dimensional differentiable manifold S^2
sage: polar(p)     # abs tol 1e-12
(2.0853981633974477, 1.4203177070475606)
sage: tgt_vec = c.tangent_vector_eval_at(1.3, verbose=True)
Evaluating tangent vector components from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: tgt_vec[:]    # abs tol 1e-12
[1.000000000000011, 1.148779968412235]

integrated_curve(equations_rhs, velocities, curve_param, initial_tangent_vector, chart=None, name=None, latex_name=None, verbose=False, across_charts=False)#

Construct a curve defined by a system of second order differential equations in the coordinate functions.

See also

IntegratedCurve for details.

INPUT:

equations_rhs – list of the right-hand sides of the equations on the velocities only
velocities – list of the symbolic expressions used in equations_rhs to denote the velocities
curve_param – a tuple of the type (t, t_min, t_max), where
- t is the symbolic variable used in equations_rhs to denote the parameter of the curve;
- t_min is its minimal (finite) value;
- t_max its maximal (finite) value.
initial_tangent_vector – TangentVector; initial tangent vector of the curve
chart – (default: None) chart on the manifold in which the equations are given; if None the default chart of the manifold is assumed
name – (default: None) string; symbol given to the curve
latex_name – (default: None) string; LaTeX symbol to denote the curve; if none is provided, name will be used

OUTPUT:

IntegratedCurve

EXAMPLES:

Trajectory of a particle of unit mass and unit charge in a unit, uniform, stationary magnetic field:

sage: M = Manifold(3, 'M')
sage: X.<x1,x2,x3> = M.chart()
sage: t = var('t')
sage: D = X.symbolic_velocities()
sage: eqns = [D[1], -D[0], SR(0)]
sage: p = M.point((0,0,0), name='p')
sage: Tp = M.tangent_space(p)
sage: v = Tp((1,0,1))
sage: c = M.integrated_curve(eqns, D, (t,0,6), v, name='c'); c
Integrated curve c in the 3-dimensional differentiable
 manifold M
sage: sys = c.system(verbose=True)
Curve c in the 3-dimensional differentiable manifold M
 integrated over the Real interval (0, 6) as a solution to
 the following system, written with respect to
 Chart (M, (x1, x2, x3)):

Initial point: Point p on the 3-dimensional differentiable
 manifold M with coordinates [0, 0, 0] with respect to
 Chart (M, (x1, x2, x3))
Initial tangent vector: Tangent vector at Point p on the
 3-dimensional differentiable manifold M with
 components [1, 0, 1] with respect to Chart (M, (x1, x2, x3))

d(x1)/dt = Dx1
d(x2)/dt = Dx2
d(x3)/dt = Dx3
d(Dx1)/dt = Dx2
d(Dx2)/dt = -Dx1
d(Dx3)/dt = 0

sage: sol = c.solve()
sage: interp = c.interpolate()
sage: p = c(1.3, verbose=True)
Evaluating point coordinates from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: p
Point on the 3-dimensional differentiable manifold M
sage: p.coordinates()     # abs tol 1e-12
(0.9635581599167499, -0.7325011788437327, 1.3)
sage: tgt_vec = c.tangent_vector_eval_at(3.7, verbose=True)
Evaluating tangent vector components from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: tgt_vec[:]    # abs tol 1e-12
[-0.8481007454066425, 0.5298350137284363, 1.0]

integrated_geodesic(metric, curve_param, initial_tangent_vector, chart=None, name=None, latex_name=None, verbose=False, across_charts=False)#

Construct a geodesic on the manifold with respect to a given metric.

See also

IntegratedGeodesic for details.

INPUT:

metric – PseudoRiemannianMetric metric with respect to which the curve is a geodesic
curve_param – a tuple of the type (t, t_min, t_max), where
- t is the symbolic variable to be used as the parameter of the curve (the equations defining an instance of IntegratedGeodesic are such that t will actually be an affine parameter of the curve);
- t_min is its minimal (finite) value;
- t_max its maximal (finite) value.
initial_tangent_vector – TangentVector; initial tangent vector of the curve
chart – (default: None) chart on the manifold in which the equations are given; if None the default chart of the manifold is assumed
name – (default: None) string; symbol given to the curve
latex_name – (default: None) string; LaTeX symbol to denote the curve; if none is provided, name will be used

OUTPUT:

IntegratedGeodesic

EXAMPLES:

Geodesics of the unit 2-sphere \(\mathbb{S}^{2}\):

sage: S2 = Manifold(2, 'S^2', start_index=1)
sage: polar.<th,ph> = S2.chart('th ph')
sage: epolar = polar.frame()

Set the standard metric tensor \(g\) on \(\mathbb{S}^{2}\):

sage: g = S2.metric('g')
sage: g[1,1], g[2,2] = 1, (sin(th))^2

Declare an integrated geodesic with respect to this metric:

sage: p = S2.point((pi/4, 0), name='p')
sage: Tp = S2.tangent_space(p)
sage: v = Tp((1, 1), basis=epolar.at(p))
sage: t = var('t')
sage: c = S2.integrated_geodesic(g, (t, 0, 6), v,
....:                                 chart=polar, name='c')
sage: sys = c.system(verbose=True)
Geodesic c in the 2-dimensional differentiable manifold S^2
 equipped with Riemannian metric g on the 2-dimensional
 differentiable manifold S^2, and integrated over the Real
 interval (0, 6) as a solution to the following geodesic
 equations, written with respect to Chart (S^2, (th, ph)):

Initial point: Point p on the 2-dimensional differentiable
manifold S^2 with coordinates [1/4*pi, 0] with respect to
Chart (S^2, (th, ph))
Initial tangent vector: Tangent vector at Point p on the
2-dimensional differentiable manifold S^2 with
components [1, 1] with respect to Chart (S^2, (th, ph))

d(th)/dt = Dth
d(ph)/dt = Dph
d(Dth)/dt = Dph^2*cos(th)*sin(th)
d(Dph)/dt = -2*Dph*Dth*cos(th)/sin(th)

sage: sol = c.solve()
sage: interp = c.interpolate()
sage: p = c(1.3, verbose=True)
Evaluating point coordinates from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: p
Point on the 2-dimensional differentiable manifold S^2
sage: p.coordinates()     # abs tol 1e-12
(2.2047435672397526, 0.7986602654406825)
sage: tgt_vec = c.tangent_vector_eval_at(3.7, verbose=True)
Evaluating tangent vector components from the interpolation
 associated with the key 'cubic spline-interp-odeint'
 by default...
sage: tgt_vec[:]    # abs tol 1e-12
[-1.0907409234671228, 0.6205670379855032]

is_manifestly_parallelizable()#

Return True if self is known to be a parallelizable and False otherwise.

If False is returned, either the manifold is not parallelizable or no vector frame has been defined on it yet.

EXAMPLES:

A just created manifold is a priori not manifestly parallelizable:

sage: M = Manifold(2, 'M')
sage: M.is_manifestly_parallelizable()
False

Defining a vector frame on it makes it parallelizable:

sage: e = M.vector_frame('e')
sage: M.is_manifestly_parallelizable()
True

Defining a coordinate chart on the whole manifold also makes it parallelizable:

sage: N = Manifold(4, 'N')
sage: X.<t,x,y,z> = N.chart()
sage: N.is_manifestly_parallelizable()
True

lorentzian_metric(name, signature='positive', latex_name=None, dest_map=None)#

Define a Lorentzian metric on the manifold.

A Lorentzian metric is a field of nondegenerate symmetric bilinear forms acting in the tangent spaces, with signature \((-,+,\cdots,+)\) or \((+,-,\cdots,-)\).

See PseudoRiemannianMetric for a complete documentation.

INPUT:

name – name given to the metric
signature – (default: ‘positive’) sign of the metric signature:
- if set to ‘positive’, the signature is n-2, where n is the manifold’s dimension, i.e. \((-,+,\cdots,+)\)
- if set to ‘negative’, the signature is -n+2, i.e. \((+,-,\cdots,-)\)
latex_name – (default: None) LaTeX symbol to denote the metric; if None, it is formed from name
dest_map – (default: None) instance of class DiffMap representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; if None, the identity map is assumed (case of a metric tensor field on \(U\))

OUTPUT:

instance of PseudoRiemannianMetric representing the defined Lorentzian metric.

EXAMPLES:

Metric of Minkowski spacetime:

sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: g = M.lorentzian_metric('g'); g
Lorentzian metric g on the 4-dimensional differentiable manifold M
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1
sage: g.display()
g = -dt⊗dt + dx⊗dx + dy⊗dy + dz⊗dz
sage: g.signature()
2

Choice of a negative signature:

sage: g = M.lorentzian_metric('g', signature='negative'); g
Lorentzian metric g on the 4-dimensional differentiable manifold M
sage: g[0,0], g[1,1], g[2,2], g[3,3] = 1, -1, -1, -1
sage: g.display()
g = dt⊗dt - dx⊗dx - dy⊗dy - dz⊗dz
sage: g.signature()
-2

metric(name, signature=None, latex_name=None, dest_map=None)#

Define a pseudo-Riemannian metric on the manifold.

A pseudo-Riemannian metric is a field of nondegenerate symmetric bilinear forms acting in the tangent spaces. See PseudoRiemannianMetric for a complete documentation.

INPUT:

name – name given to the metric
signature – (default: None) signature \(S\) of the metric as a single integer: \(S = n_+ - n_-\), where \(n_+\) (resp. \(n_-\)) is the number of positive terms (resp. number of negative terms) in any diagonal writing of the metric components; if signature is not provided, \(S\) is set to the manifold’s dimension (Riemannian signature)
latex_name – (default: None) LaTeX symbol to denote the metric; if None, it is formed from name
dest_map – (default: None) instance of class DiffMap representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; if None, the identity map is assumed (case of a metric tensor field on \(U\))

OUTPUT:

instance of PseudoRiemannianMetric representing the defined pseudo-Riemannian metric.

EXAMPLES:

Metric on a 3-dimensional manifold:

sage: M = Manifold(3, 'M', start_index=1)
sage: c_xyz.<x,y,z> = M.chart()
sage: g = M.metric('g'); g
Riemannian metric g on the 3-dimensional differentiable manifold M

See also

PseudoRiemannianMetric for more examples.

mixed_form(comp=None, name=None, latex_name=None, dest_map=None)#

Define a mixed form on self.

Via the argument dest_map, it is possible to let the mixed form take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold, \(\Phi:\ M \rightarrow N\) a differentiable map, a mixed form along \(\Phi\) can be considered as a differentiable map

\[a: M \longrightarrow \bigoplus^n_{k=0} T^{(0,k)}N\]

(\(T^{(0,k)} N\) being the tensor bundle of type \((0,k)\) over \(N\), \(\oplus\) being the Whitney sum and \(n\) being the dimension of \(N\)) such that

\[\forall x \in M,\quad a(x) \in \bigoplus^n_{k=0} \Lambda^k(T^*_{\Phi(x)} N),\]

where \(\Lambda^k(T^*_{\Phi(x)} N)\) is the \(k\)-th exterior power of the dual of the tangent space \(T_{\Phi(x)} N\).

The standard case of a mixed form on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\).

See also

MixedForm for complete documentation.

INPUT:

comp – (default: None) homogeneous components of the mixed form as a list; if none is provided, the components are set to innocent unnamed differential forms
name – (default: None) name given to the differential form
latex_name – (default: None) LaTeX symbol to denote the differential form; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a differential form on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

the mixed form as a MixedForm

EXAMPLES:

A mixed form on an open subset of a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: U = M.open_subset('U', latex_name=r'\mathcal{U}'); U
Open subset U of the 3-dimensional differentiable manifold M
sage: c_xyz.<x,y,z> = U.chart()
sage: f = U.mixed_form(name='F'); f
Mixed differential form F on the Open subset U of the 3-dimensional
 differentiable manifold M

See the documentation of class MixedForm for more examples.

mixed_form_algebra(dest_map=None)#

Return the set of mixed forms defined on self, possibly with values in another manifold, as a graded algebra.

See also

MixedFormAlgebra for complete documentation.

INPUT:

dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of mixed forms on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a MixedFormAlgebra representing the graded algebra \(\Omega^*(M,\Phi)\) of mixed forms on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Graded algebra of mixed forms on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: M.mixed_form_algebra()
Graded algebra Omega^*(M) of mixed differential forms on the
 2-dimensional differentiable manifold M
sage: M.mixed_form_algebra().category()
Join of Category of graded algebras over Symbolic Ring and Category of chain complexes over Symbolic Ring
sage: M.mixed_form_algebra().base_ring()
Symbolic Ring

The outcome is cached:

sage: M.mixed_form_algebra() is M.mixed_form_algebra()
True

multivector_field(*args, **kwargs)#

Define a multivector field on self.

Via the argument dest_map, it is possible to let the multivector field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold, \(\Phi:\ M \rightarrow N\) a differentiable map and \(p\) a non-negative integer, a multivector field of degree \(p\) (or \(p\)-vector field) along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(p,0)} N\]

(\(T^{(p,0)} N\) being the tensor bundle of type \((p,0)\) over \(N\)) such that

\[\forall x \in M,\quad t(x) \in \Lambda^p(T_{\Phi(x)} N),\]

where \(\Lambda^p(T_{\Phi(x)} N)\) is the \(p\)-th exterior power of the tangent vector space \(T_{\Phi(x)} N\).

The standard case of a \(p\)-vector field on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

For \(p = 1\), one can use the method vector_field() instead.

See also

MultivectorField and MultivectorFieldParal for a complete documentation.

INPUT:

degree – the degree \(p\) of the multivector field (i.e. its tensor rank)
comp – (optional) either the components of the multivector field with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the multivector field
latex_name – (default: None) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a multivector field on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

the \(p\)-vector field as a MultivectorField (or if \(N\) is parallelizable, a MultivectorFieldParal)

EXAMPLES:

A 2-vector field on a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: h = M.multivector_field(2, name='H'); h
2-vector field H on the 3-dimensional differentiable manifold M
sage: h[0,1], h[0,2], h[1,2] = x+y, x*z, -3
sage: h.display()
H = (x + y) ∂/∂x∧∂/∂y + x*z ∂/∂x∧∂/∂z - 3 ∂/∂y∧∂/∂z

For more examples, see MultivectorField and MultivectorFieldParal.

multivector_module(degree, dest_map=None)#

Return the set of multivector fields of a given degree defined on self, possibly with values in another manifold, as a module over the algebra of scalar fields defined on self.

See also

MultivectorModule for complete documentation.

INPUT:

degree – positive integer; the degree \(p\) of the multivector fields
dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of multivector fields on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a MultivectorModule (or if \(N\) is parallelizable, a MultivectorFreeModule) representing the module \(\Omega^p(M,\Phi)\) of \(p\)-forms on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Module of 2-vector fields on a 3-dimensional parallelizable manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: M.multivector_module(2)
Free module A^2(M) of 2-vector fields on the 3-dimensional
 differentiable manifold M
sage: M.multivector_module(2).category()
Category of finite dimensional modules over Algebra of
 differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: M.multivector_module(2).base_ring()
Algebra of differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: M.multivector_module(2).rank()
3

The outcome is cached:

sage: M.multivector_module(2) is M.multivector_module(2)
True

one_form(*comp, **kwargs)#

Define a 1-form on the manifold.

Via the argument dest_map, it is possible to let the 1-form take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold and \(\Phi:\ M \rightarrow N\) a differentiable map, a 1-form along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^* N\]

(\(T^* N\) being the cotangent bundle of \(N\)) such that

\[\forall p \in M,\quad t(p) \in T^*_{\Phi(p)}N,\]

where \(T^*_{\Phi(p)}\) is the dual of the tangent space \(T_{\Phi(p)} N\).

The standard case of a 1-form on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

See also

DiffForm and DiffFormParal for a complete documentation.

INPUT:

comp – (optional) either the components of 1-form with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the 1-form
latex_name – (default: None) LaTeX symbol to denote the 1-form; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a 1-form on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

the 1-form as a DiffForm (or if \(N\) is parallelizable, a DiffFormParal)

EXAMPLES:

A 1-form on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: om = M.one_form(-y, 2+x, name='omega', latex_name=r'\omega')
sage: om
1-form omega on the 2-dimensional differentiable manifold M
sage: om.display()
omega = -y dx + (x + 2) dy
sage: om.parent()
Free module Omega^1(M) of 1-forms on the 2-dimensional
 differentiable manifold M

For more examples, see DiffForm and DiffFormParal.

open_subset(name, latex_name=None, coord_def={}, supersets=None)#

Create an open subset of the manifold.

An open subset is a set that is (i) included in the manifold and (ii) open with respect to the manifold’s topology. It is a differentiable manifold by itself. Hence the returned object is an instance of DifferentiableManifold.

INPUT:

name – name given to the open subset
latex_name – (default: None) LaTeX symbol to denote the subset; if none is provided, it is set to name
coord_def – (default: {}) definition of the subset in terms of coordinates; coord_def must a be dictionary with keys charts in the manifold’s atlas and values the symbolic expressions formed by the coordinates to define the subset.
supersets – (default: only self) list of sets that the new open subset is a subset of

OUTPUT:

the open subset, as an instance of DifferentiableManifold

EXAMPLES:

Creating an open subset of a differentiable manifold:

sage: M = Manifold(2, 'M')
sage: A = M.open_subset('A'); A
Open subset A of the 2-dimensional differentiable manifold M

As an open subset of a differentiable manifold, A is itself a differentiable manifold, on the same topological field and of the same dimension as M:

sage: A.category()
Join of Category of subobjects of sets and Category of smooth
 manifolds over Real Field with 53 bits of precision
sage: A.base_field() == M.base_field()
True
sage: dim(A) == dim(M)
True

Creating an open subset of A:

sage: B = A.open_subset('B'); B
Open subset B of the 2-dimensional differentiable manifold M

We have then:

sage: A.subset_family()
Set {A, B} of open subsets of the 2-dimensional differentiable manifold M
sage: B.is_subset(A)
True
sage: B.is_subset(M)
True

Defining an open subset by some coordinate restrictions: the open unit disk in of the Euclidean plane:

sage: X.<x,y> = M.chart() # Cartesian coordinates on M
sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1}); U
Open subset U of the 2-dimensional differentiable manifold M

Since the argument coord_def has been set, U is automatically endowed with a chart, which is the restriction of X to U:

sage: U.atlas()
[Chart (U, (x, y))]
sage: U.default_chart()
Chart (U, (x, y))
sage: U.default_chart() is X.restrict(U)
True

A point in U:

sage: p = U.an_element(); p
Point on the 2-dimensional differentiable manifold M
sage: X(p)  # the coordinates (x,y) of p
(0, 0)
sage: p in U
True

Checking whether various points, defined by their coordinates with respect to chart X, are in U:

sage: M((0,1/2)) in U
True
sage: M((0,1)) in U
False
sage: M((1/2,1)) in U
False
sage: M((-1/2,1/3)) in U
True

orientation()#

Get the preferred orientation of self if available.

An orientation on a differentiable manifold is an atlas of charts whose transition maps are pairwise orientation preserving, i.e. whose Jacobian determinants are pairwise positive.

A differentiable manifold with an orientation is called orientable.

A differentiable manifold is orientable if and only if the tangent bundle is orientable in terms of a vector bundle, see orientation().

Note

In contrast to topological manifolds, see orientation(), differentiable manifolds preferably use the notion of orientability in terms of the tangent bundle.

The trivial case corresponds to the manifold being parallelizable, i.e. admitting a frame covering the whole manifold. In that case, if no preferred orientation has been manually set before, one of those frames (usually the default frame) is set to the preferred orientation on self and returned here.

EXAMPLES:

In case one frame already covers the manifold, an orientation is readily obtained:

sage: M = Manifold(3, 'M')
sage: c.<x,y,z> = M.chart()
sage: M.orientation()
[Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z))]

However, orientations are usually not easy to obtain:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: M.orientation()
[]

In that case, the orientation can be set by the user; either in terms of charts or in terms of frames:

sage: M.set_orientation([c_xy, c_uv])
sage: M.orientation()
[Coordinate frame (U, (∂/∂x,∂/∂y)),
 Coordinate frame (V, (∂/∂u,∂/∂v))]
sage: M.set_orientation([c_xy.frame(), c_uv.frame()])
sage: M.orientation()
[Coordinate frame (U, (∂/∂x,∂/∂y)),
 Coordinate frame (V, (∂/∂u,∂/∂v))]

The orientation on submanifolds are inherited from the ambient manifold:

sage: W = U.intersection(V, name='W')
sage: W.orientation()
[Vector frame (W, (∂/∂x,∂/∂y))]

poisson_tensor(name=None, latex_name=None)#

Construct a Poisson tensor on the current manifold.

OUTPUT:

instance of PoissonTensorField

EXAMPLES:

Standard Poisson tensor on \(\RR^2\):

sage: M.<q, p> = EuclideanSpace(2)
sage: poisson = M.poisson_tensor('varpi')
sage: poisson.set_comp()[1,2] = -1
sage: poisson.display()
varpi = -e_q∧e_p

riemannian_metric(name, latex_name=None, dest_map=None)#

Define a Riemannian metric on the manifold.

A Riemannian metric is a field of positive definite symmetric bilinear forms acting in the tangent spaces.

See PseudoRiemannianMetric for a complete documentation.

INPUT:

name – name given to the metric
latex_name – (default: None) LaTeX symbol to denote the metric; if None, it is formed from name
dest_map – (default: None) instance of class DiffMap representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; if None, the identity map is assumed (case of a metric tensor field on \(U\))

OUTPUT:

instance of PseudoRiemannianMetric representing the defined Riemannian metric.

EXAMPLES:

Metric of the hyperbolic plane \(H^2\):

sage: H2 = Manifold(2, 'H^2', start_index=1)
sage: X.<x,y> = H2.chart('x y:(0,+oo)')  # Poincaré half-plane coord.
sage: g = H2.riemannian_metric('g')
sage: g[1,1], g[2,2] = 1/y^2, 1/y^2
sage: g
Riemannian metric g on the 2-dimensional differentiable manifold H^2
sage: g.display()
g = y^(-2) dx⊗dx + y^(-2) dy⊗dy
sage: g.signature()
2

See also

PseudoRiemannianMetric for more examples.

set_change_of_frame(frame1, frame2, change_of_frame, compute_inverse=True)#

Relate two vector frames by an automorphism.

This updates the internal dictionary self._frame_changes.

INPUT:

frame1 – frame 1, denoted \((e_i)\) below
frame2 – frame 2, denoted \((f_i)\) below
change_of_frame – instance of class AutomorphismFieldParal describing the automorphism \(P\) that relates the basis \((e_i)\) to the basis \((f_i)\) according to \(f_i = P(e_i)\)
compute_inverse (default: True) – if set to True, the inverse automorphism is computed and the change from basis \((f_i)\) to \((e_i)\) is set to it in the internal dictionary self._frame_changes

EXAMPLES:

Connecting two vector frames on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: e = M.vector_frame('e')
sage: f = M.vector_frame('f')
sage: a = M.automorphism_field()
sage: a[e,:] = [[1,2],[0,3]]
sage: M.set_change_of_frame(e, f, a)
sage: f[0].display(e)
f_0 = e_0
sage: f[1].display(e)
f_1 = 2 e_0 + 3 e_1
sage: e[0].display(f)
e_0 = f_0
sage: e[1].display(f)
e_1 = -2/3 f_0 + 1/3 f_1
sage: M.change_of_frame(e,f)[e,:]
[1 2]
[0 3]

set_default_frame(frame)#

Changing the default vector frame on self.

INPUT:

frame – VectorFrame a vector frame defined on some subset of self

EXAMPLES:

Changing the default frame on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: e = M.vector_frame('e')
sage: M.default_frame()
Coordinate frame (M, (∂/∂x,∂/∂y))
sage: M.set_default_frame(e)
sage: M.default_frame()
Vector frame (M, (e_0,e_1))

set_orientation(orientation)#

Set the preferred orientation of self.

INPUT:

orientation – either a chart / list of charts, or a vector frame / list of vector frames, covering self

Warning

It is the user’s responsibility that the orientation set here is indeed an orientation. There is no check going on in the background. See orientation() for the definition of an orientation.

EXAMPLES:

Set an orientation on a manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart(); c_uv.<u,v> = M.chart()
sage: M.set_orientation(c_uv)
sage: M.orientation()
[Coordinate frame (M, (∂/∂u,∂/∂v))]

Instead of a chart, a vector frame can be given, too:

sage: M.set_orientation(c_xy.frame())
sage: M.orientation()
[Coordinate frame (M, (∂/∂x,∂/∂y))]

Set an orientation in the non-trivial case:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: M.set_orientation([c_xy, c_uv])
sage: M.orientation()
[Coordinate frame (U, (∂/∂x,∂/∂y)),
 Coordinate frame (V, (∂/∂u,∂/∂v))]

Again, the vector frame notion can be used instead:

sage: M.set_orientation([c_xy.frame(), c_uv.frame()])
sage: M.orientation()
[Coordinate frame (U, (∂/∂x,∂/∂y)),
 Coordinate frame (V, (∂/∂u,∂/∂v))]

sym_bilin_form_field(*comp, **kwargs)#

Define a field of symmetric bilinear forms on self.

Via the argument dest_map, it is possible to let the field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold and \(\Phi:\ M \rightarrow N\) a differentiable map, a field of symmetric bilinear forms along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(0,2)}N\]

(\(T^{(0,2)} N\) being the tensor bundle of type \((0,2)\) over \(N\)) such that

\[\forall p \in M,\ t(p) \in S(T_{\Phi(p)} N),\]

where \(S(T_{\Phi(p)} N)\) is the space of symmetric bilinear forms on the tangent space \(T_{\Phi(p)} N\).

The standard case of fields of symmetric bilinear forms on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

INPUT:

comp – (optional) either the components of the field of symmetric bilinear forms with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the field
latex_name – (default: None) LaTeX symbol to denote the field; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field on \(M\)), otherwise dest_map must be an instance of instance of class DiffMap

OUTPUT:

a TensorField (or if \(N\) is parallelizable, a TensorFieldParal) of tensor type \((0,2)\) and symmetric representing the defined field of symmetric bilinear forms

EXAMPLES:

A field of symmetric bilinear forms on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: t = M.sym_bilin_form_field(name='T'); t
Field of symmetric bilinear forms T on the 2-dimensional
 differentiable manifold M

Such a object is a tensor field of rank 2 and type \((0,2)\):

sage: t.parent()
Free module T^(0,2)(M) of type-(0,2) tensors fields on the
 2-dimensional differentiable manifold M
sage: t.tensor_rank()
2
sage: t.tensor_type()
(0, 2)

The LaTeX symbol is deduced from the name or can be specified when creating the object:

sage: latex(t)
T
sage: om = M.sym_bilin_form_field(name='Omega', latex_name=r'\Omega')
sage: latex(om)
\Omega

Setting the components in the manifold’s default vector frame:

sage: t[0,0], t[0,1], t[1,1] = -1, x, x*y

The unset components are either zero or deduced by symmetry:

sage: t[1, 0]
x
sage: t[:]
[ -1   x]
[  x x*y]

One can also set the components while defining the field of symmetric bilinear forms:

sage: t = M.sym_bilin_form_field([[-1, x], [x, x*y]], name='T')

A symmetric bilinear form acts on vector pairs:

sage: v1 = M.vector_field(y, x, name='V_1')
sage: v2 = M.vector_field(x+y, 2, name='V_2')
sage: s = t(v1,v2) ; s
Scalar field T(V_1,V_2) on the 2-dimensional differentiable
 manifold M
sage: s.expr()
x^3 + (3*x^2 + x)*y - y^2
sage: s.expr() - t[0,0]*v1[0]*v2[0] - \
....: t[0,1]*(v1[0]*v2[1]+v1[1]*v2[0]) - t[1,1]*v1[1]*v2[1]
0
sage: latex(s)
T\left(V_1,V_2\right)

Adding two symmetric bilinear forms results in another symmetric bilinear form:

sage: a = M.sym_bilin_form_field([[1, 2], [2, 3]])
sage: b = M.sym_bilin_form_field([[-1, 4], [4, 5]])
sage: s = a + b ; s
Field of symmetric bilinear forms on the 2-dimensional
 differentiable manifold M
sage: s[:]
[0 6]
[6 8]

But adding a symmetric bilinear from with a non-symmetric bilinear form results in a generic type \((0,2)\) tensor:

sage: c = M.tensor_field(0, 2, [[-2, -3], [1,7]])
sage: s1 = a + c ; s1
Tensor field of type (0,2) on the 2-dimensional differentiable
 manifold M
sage: s1[:]
[-1 -1]
[ 3 10]
sage: s2 = c + a ; s2
Tensor field of type (0,2) on the 2-dimensional differentiable
 manifold M
sage: s2[:]
[-1 -1]
[ 3 10]

symplectic_form(name=None, latex_name=None)#

Construct a symplectic form on the current manifold.

OUTPUT:

instance of SymplecticForm

EXAMPLES:

Standard symplectic form on \(\RR^2\):

sage: M.<q, p> = EuclideanSpace(2)
sage: omega = M.symplectic_form('omega', r'\omega')
sage: omega.set_comp()[1,2] = -1
sage: omega.display()
omega = -dq∧dp

tangent_bundle(dest_map=None)#

Return the tangent bundle possibly along a destination map with base space self.

See also

TensorBundle for complete documentation.

INPUT:

dest_map – (default: None) destination map \(\Phi:\ M \rightarrow N\) (type: DiffMap) from which the tangent bundle is pulled back; if None, it is assumed that \(N=M\) and \(\Phi\) is the identity map of \(M\) (case of the standard tangent bundle over \(M\))

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: TM = M.tangent_bundle(); TM
Tangent bundle TM over the 2-dimensional differentiable manifold M

tangent_identity_field(dest_map=None)#

Return the field of identity maps in the tangent spaces on self.

Via the argument dest_map, it is possible to let the field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold and \(\Phi:\ M \rightarrow N\) a differentiable map, a field of identity maps along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(1,1)} N\]

(\(T^{(1,1)} N\) being the tensor bundle of type \((1,1)\) over \(N\)) such that

\[\forall p \in M,\ t(p) = \mathrm{Id}_{T_{\Phi(p)} N},\]

where \(\mathrm{Id}_{T_{\Phi(p)} N}\) is the identity map of the tangent space \(T_{\Phi(p)} N\).

The standard case of a field of identity maps on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

INPUT:

name – (string; default: ‘Id’) name given to the field of identity maps
latex_name – (string; default: None) LaTeX symbol to denote the field of identity map; if none is provided, the LaTeX symbol is set to ‘mathrm{Id}’ if name is ‘Id’ and to name otherwise
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field of identity maps on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a AutomorphismField (or if \(N\) is parallelizable, a AutomorphismFieldParal) representing the field of identity maps

EXAMPLES:

Field of tangent-space identity maps on a 3-dimensional manifold:

sage: M = Manifold(3, 'M', start_index=1)
sage: c_xyz.<x,y,z> = M.chart()
sage: a = M.tangent_identity_field(); a
Field of tangent-space identity maps on the 3-dimensional
 differentiable manifold M
sage: a.comp()
Kronecker delta of size 3x3

For more examples, see AutomorphismField.

tangent_space(point, base_ring=None)#

Tangent space to self at a given point.

INPUT:

point – ManifoldPoint; point \(p\) on the manifold
base_ring – (default: the symbolic ring) the base ring

OUTPUT:

TangentSpace representing the tangent vector space \(T_{p} M\), where \(M\) is the current manifold

EXAMPLES:

A tangent space to a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: p = M.point((2, -3), name='p')
sage: Tp = M.tangent_space(p); Tp
Tangent space at Point p on the 2-dimensional differentiable
 manifold M
sage: Tp.category()
Category of finite dimensional vector spaces over Symbolic Ring
sage: dim(Tp)
2

See also

TangentSpace for more examples.

tangent_vector(*args, **kwargs)#

Define a tangent vector at a given point of self.

INPUT:

point – ManifoldPoint; point \(p\) on self
comp – components of the vector with respect to the basis specified by the argument basis, either as an iterable or as a sequence of \(n\) components, \(n\) being the dimension of self (see examples below)
basis – (default: None) FreeModuleBasis; basis of the tangent space at \(p\) with respect to which the components are defined; if None, the default basis of the tangent space is used
name – (default: None) string; symbol given to the vector
latex_name – (default: None) string; LaTeX symbol to denote the vector; if None, name will be used

OUTPUT:

TangentVector representing the tangent vector at point \(p\)

EXAMPLES:

Vector at a point \(p\) of the Euclidean plane:

sage: E.<x,y>= EuclideanSpace()
sage: p = E((1, 2), name='p')
sage: v = E.tangent_vector(p, -1, 3, name='v'); v
Vector v at Point p on the Euclidean plane E^2
sage: v.display()
v = -e_x + 3 e_y
sage: v.parent()
Tangent space at Point p on the Euclidean plane E^2
sage: v in E.tangent_space(p)
True

An alias of tangent_vector is vector:

sage: v = E.vector(p, -1, 3, name='v'); v
Vector v at Point p on the Euclidean plane E^2

The components can be passed as a tuple or a list:

sage: v1 = E.vector(p, (-1, 3)); v1
Vector at Point p on the Euclidean plane E^2
sage: v1 == v
True

or as an object created by the vector function:

sage: v2 = E.vector(p, vector([-1, 3])); v2
Vector at Point p on the Euclidean plane E^2
sage: v2 == v
True

Example of use with the options basis and latex_name:

sage: polar_basis = E.polar_frame().at(p)
sage: polar_basis
Basis (e_r,e_ph) on the Tangent space at Point p on the Euclidean plane E^2
sage: v = E.vector(p, 2, -1, basis=polar_basis, name='v',
....:              latex_name=r'\vec{v}')
sage: v
Vector v at Point p on the Euclidean plane E^2
sage: v.display(polar_basis)
v = 2 e_r - e_ph
sage: v.display()
v = 4/5*sqrt(5) e_x + 3/5*sqrt(5) e_y
sage: latex(v)
\vec{v}

tensor_bundle(k, l, dest_map=None)#

Return a tensor bundle of type \((k, l)\) defined over self, possibly along a destination map.

INPUT:

k – the contravariant rank of the tensor bundle
l – the covariant rank of the tensor bundle
dest_map – (default: None) destination map \(\Phi:\ M \rightarrow N\) (type: DiffMap) from which the tensor bundle is pulled back; if None, it is assumed that \(N=M\) and \(\Phi\) is the identity map of \(M\) (case of the standard tangent bundle over \(M\))

OUTPUT:

a TensorBundle representing a tensor bundle of type-\((k,l)\) over self

EXAMPLES:

A tensor bundle over a parallelizable 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()  # makes M parallelizable
sage: M.tensor_bundle(1, 2)
Tensor bundle T^(1,2)M over the 2-dimensional differentiable
 manifold M

The special case of the tangent bundle as tensor bundle of type (1,0):

sage: M.tensor_bundle(1,0)
Tangent bundle TM over the 2-dimensional differentiable manifold M

The result is cached:

sage: M.tensor_bundle(1, 2) is M.tensor_bundle(1, 2)
True

See also

TensorBundle for more examples and documentation.

tensor_field(*args, **kwargs)#

Define a tensor field on self.

Via the argument dest_map, it is possible to let the tensor field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold, \(\Phi:\ M \rightarrow N\) a differentiable map and \((k,l)\) a pair of non-negative integers, a tensor field of type \((k,l)\) along \(M\) with values on \(N\) is a differentiable map

\[t:\ M \longrightarrow T^{(k,l)} N\]

(\(T^{(k,l)}N\) being the tensor bundle of type \((k,l)\) over \(N\)) such that

\[\forall p \in M,\ t(p) \in T^{(k,l)}(T_{\Phi(p)} N),\]

where \(T^{(k,l)}(T_{\Phi(p)} N)\) is the space of tensors of type \((k,l)\) on the tangent space \(T_{\Phi(p)} N\).

The standard case of tensor fields on \(M\) corresponds to \(N=M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

See also

TensorField and TensorFieldParal for a complete documentation.

INPUT:

k – the contravariant rank \(k\), the tensor type being \((k,l)\)
l – the covariant rank \(l\), the tensor type being \((k,l)\)
comp – (optional) either the components of the tensor field with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the tensor field
latex_name – (default: None) LaTeX symbol to denote the tensor field; if None, the LaTeX symbol is set to name
sym – (default: None) a symmetry or a list of symmetries among the tensor arguments: each symmetry is described by a tuple containing the positions of the involved arguments, with the convention position=0 for the first argument; for instance:
- sym = (0,1) for a symmetry between the 1st and 2nd arguments
- sym = [(0,2), (1,3,4)] for a symmetry between the 1st and 3rd arguments and a symmetry between the 2nd, 4th and 5th arguments
antisym – (default: None) antisymmetry or list of antisymmetries among the arguments, with the same convention as for sym
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a tensor field on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a TensorField (or if \(N\) is parallelizable, a TensorFieldParal) representing the defined tensor field

EXAMPLES:

A tensor field of type \((2,0)\) on a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: t = M.tensor_field(2, 0, [[1+x, -y], [0, x*y]], name='T'); t
Tensor field T of type (2,0) on the 2-dimensional differentiable
 manifold M
sage: t.display()
T = (x + 1) ∂/∂x⊗∂/∂x - y ∂/∂x⊗∂/∂y + x*y ∂/∂y⊗∂/∂y

The type \((2,0)\) tensor fields on \(M\) form the set \(\mathcal{T}^{(2,0)}(M)\), which is a module over the algebra \(C^k(M)\) of differentiable scalar fields on \(M\):

sage: t.parent()
Free module T^(2,0)(M) of type-(2,0) tensors fields on the
 2-dimensional differentiable manifold M
sage: t in M.tensor_field_module((2,0))
True

For more examples, see TensorField and TensorFieldParal.

tensor_field_module(tensor_type, dest_map=None)#

Return the set of tensor fields of a given type defined on self, possibly with values in another manifold, as a module over the algebra of scalar fields defined on self.

See also

TensorFieldModule for a complete documentation.

INPUT:

tensor_type – pair \((k,l)\) with \(k\) being the contravariant rank and \(l\) the covariant rank
dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of tensor fields on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a TensorFieldModule (or if \(N\) is parallelizable, a TensorFieldFreeModule) representing the module \(\mathcal{T}^{(k,l)}(M,\Phi)\) of type-\((k,l)\) tensor fields on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Module of type-\((2,1)\) tensor fields on a 3-dimensional open subset of a differentiable manifold:

sage: M = Manifold(3, 'M')
sage: U = M.open_subset('U')
sage: c_xyz.<x,y,z> = U.chart()
sage: TU = U.tensor_field_module((2,1)) ; TU
Free module T^(2,1)(U) of type-(2,1) tensors fields on the Open
 subset U of the 3-dimensional differentiable manifold M
sage: TU.category()
Category of tensor products of finite dimensional modules
 over Algebra of differentiable scalar fields
  on the Open subset U of the 3-dimensional differentiable manifold M
sage: TU.base_ring()
Algebra of differentiable scalar fields on the Open subset U of
 the 3-dimensional differentiable manifold M
sage: TU.base_ring() is U.scalar_field_algebra()
True
sage: TU.an_element()
Tensor field of type (2,1) on the Open subset U of the
 3-dimensional differentiable manifold M
sage: TU.an_element().display()
2 ∂/∂x⊗∂/∂x⊗dx

vector(*args, **kwargs)#

Define a tangent vector at a given point of self.

INPUT:

point – ManifoldPoint; point \(p\) on self
comp – components of the vector with respect to the basis specified by the argument basis, either as an iterable or as a sequence of \(n\) components, \(n\) being the dimension of self (see examples below)
basis – (default: None) FreeModuleBasis; basis of the tangent space at \(p\) with respect to which the components are defined; if None, the default basis of the tangent space is used
name – (default: None) string; symbol given to the vector
latex_name – (default: None) string; LaTeX symbol to denote the vector; if None, name will be used

OUTPUT:

TangentVector representing the tangent vector at point \(p\)

EXAMPLES:

Vector at a point \(p\) of the Euclidean plane:

sage: E.<x,y>= EuclideanSpace()
sage: p = E((1, 2), name='p')
sage: v = E.tangent_vector(p, -1, 3, name='v'); v
Vector v at Point p on the Euclidean plane E^2
sage: v.display()
v = -e_x + 3 e_y
sage: v.parent()
Tangent space at Point p on the Euclidean plane E^2
sage: v in E.tangent_space(p)
True

An alias of tangent_vector is vector:

sage: v = E.vector(p, -1, 3, name='v'); v
Vector v at Point p on the Euclidean plane E^2

The components can be passed as a tuple or a list:

sage: v1 = E.vector(p, (-1, 3)); v1
Vector at Point p on the Euclidean plane E^2
sage: v1 == v
True

or as an object created by the vector function:

sage: v2 = E.vector(p, vector([-1, 3])); v2
Vector at Point p on the Euclidean plane E^2
sage: v2 == v
True

Example of use with the options basis and latex_name:

sage: polar_basis = E.polar_frame().at(p)
sage: polar_basis
Basis (e_r,e_ph) on the Tangent space at Point p on the Euclidean plane E^2
sage: v = E.vector(p, 2, -1, basis=polar_basis, name='v',
....:              latex_name=r'\vec{v}')
sage: v
Vector v at Point p on the Euclidean plane E^2
sage: v.display(polar_basis)
v = 2 e_r - e_ph
sage: v.display()
v = 4/5*sqrt(5) e_x + 3/5*sqrt(5) e_y
sage: latex(v)
\vec{v}

vector_bundle(rank, name, field='real', latex_name=None)#

Return a differentiable vector bundle over the given field with given rank over this differentiable manifold of the same differentiability class as the manifold.

INPUT:

rank – rank of the vector bundle
name – name given to the total space
field – (default: 'real') topological field giving the vector space structure to the fibers
latex_name – optional LaTeX name for the total space

OUTPUT:

a differentiable vector bundle as an instance of DifferentiableVectorBundle

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: M.vector_bundle(2, 'E')
Differentiable real vector bundle E -> M of rank 2 over the base
 space 2-dimensional differentiable manifold M

vector_field(*comp, **kwargs)#

Define a vector field on self.

Via the argument dest_map, it is possible to let the vector field take its values on another manifold. More precisely, if \(M\) is the current manifold, \(N\) a differentiable manifold and \(\Phi:\ M \rightarrow N\) a differentiable map, a vector field along \(M\) with values on \(N\) is a differentiable map

\[v:\ M \longrightarrow TN\]

(\(TN\) being the tangent bundle of \(N\)) such that

\[\forall p \in M,\ v(p) \in T_{\Phi(p)} N,\]

where \(T_{\Phi(p)} N\) is the tangent space to \(N\) at the point \(\Phi(p)\).

The standard case of vector fields on \(M\) corresponds to \(N = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

See also

VectorField and VectorFieldParal for a complete documentation.

INPUT:

comp – (optional) either the components of the vector field with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed
frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed
chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed
name – (default: None) name given to the vector field
latex_name – (default: None) LaTeX symbol to denote the vector field; if none is provided, the LaTeX symbol is set to name
dest_map – (default: None) the destination map \(\Phi:\ M \rightarrow N\); if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a vector field on \(M\)), otherwise dest_map must be a DiffMap

OUTPUT:

a VectorField (or if \(N\) is parallelizable, a VectorFieldParal) representing the defined vector field

EXAMPLES:

A vector field on a open subset of a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: U = M.open_subset('U')
sage: c_xyz.<x,y,z> = U.chart()
sage: v = U.vector_field(y, -x*z, 1+y, name='v'); v
Vector field v on the Open subset U of the 3-dimensional
 differentiable manifold M
sage: v.display()
v = y ∂/∂x - x*z ∂/∂y + (y + 1) ∂/∂z

The vector fields on \(U\) form the set \(\mathfrak{X}(U)\), which is a module over the algebra \(C^k(U)\) of differentiable scalar fields on \(U\):

sage: v.parent()
Free module X(U) of vector fields on the Open subset U of the
 3-dimensional differentiable manifold M
sage: v in U.vector_field_module()
True

For more examples, see VectorField and VectorFieldParal.

vector_field_module(dest_map=None, force_free=False)#

Return the set of vector fields defined on self, possibly with values in another differentiable manifold, as a module over the algebra of scalar fields defined on the manifold.

See VectorFieldModule for a complete documentation.

INPUT:

dest_map – (default: None) destination map, i.e. a differentiable map \(\Phi:\ M \rightarrow N\), where \(M\) is the current manifold and \(N\) a differentiable manifold; if None, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of vector fields on \(M\)), otherwise dest_map must be a DiffMap
force_free – (default: False) if set to True, force the construction of a free module (this implies that \(N\) is parallelizable)

OUTPUT:

a VectorFieldModule (or if \(N\) is parallelizable, a VectorFieldFreeModule) representing the \(C^k(M)\)-module \(\mathfrak{X}(M,\Phi)\) of vector fields on \(M\) taking values on \(\Phi(M)\subset N\)

EXAMPLES:

Vector field module \(\mathfrak{X}(U) := \mathfrak{X}(U,\mathrm{Id}_U)\) of the complement \(U\) of the two poles on the sphere \(\mathbb{S}^2\):

sage: S2 = Manifold(2, 'S^2')
sage: U = S2.open_subset('U')  # the complement of the two poles
sage: spher_coord.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coordinates
sage: XU = U.vector_field_module() ; XU
Free module X(U) of vector fields on the Open subset U of
 the 2-dimensional differentiable manifold S^2
sage: XU.category()
Category of finite dimensional modules over Algebra of
 differentiable scalar fields on the Open subset U of
 the 2-dimensional differentiable manifold S^2
sage: XU.base_ring()
Algebra of differentiable scalar fields on the Open subset U of
 the 2-dimensional differentiable manifold S^2
sage: XU.base_ring() is U.scalar_field_algebra()
True

\(\mathfrak{X}(U)\) is a free module because \(U\) is parallelizable (being a chart domain):

sage: U.is_manifestly_parallelizable()
True

Its rank is the manifold’s dimension:

sage: XU.rank()
2

The elements of \(\mathfrak{X}(U)\) are vector fields on \(U\):

sage: XU.an_element()
Vector field on the Open subset U of the 2-dimensional
 differentiable manifold S^2
sage: XU.an_element().display()
2 ∂/∂th + 2 ∂/∂ph

Vector field module \(\mathfrak{X}(U,\Phi)\) of the \(\RR^3\)-valued vector fields along \(U\), associated with the embedding \(\Phi\) of \(\mathbb{S}^2\) into \(\RR^3\):

sage: R3 = Manifold(3, 'R^3')
sage: cart_coord.<x, y, z> = R3.chart()
sage: Phi = U.diff_map(R3,
....:      [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)], name='Phi')
sage: XU_R3 = U.vector_field_module(dest_map=Phi) ; XU_R3
Free module X(U,Phi) of vector fields along the Open subset U of
 the 2-dimensional differentiable manifold S^2 mapped into the
 3-dimensional differentiable manifold R^3
sage: XU_R3.base_ring()
Algebra of differentiable scalar fields on the Open subset U of the
 2-dimensional differentiable manifold S^2

\(\mathfrak{X}(U,\Phi)\) is a free module because \(\RR^3\) is parallelizable and its rank is 3:

sage: XU_R3.rank()
3

Without any information on the manifold, the vector field module is not free by default:

sage: M = Manifold(2, 'M')
sage: XM = M.vector_field_module()
sage: isinstance(XM, FiniteRankFreeModule)
False

In particular, declaring a coordinate chart on M would yield an error:

sage: X.<x,y> = M.chart()
Traceback (most recent call last):
...
ValueError: the Module X(M) of vector fields on the 2-dimensional
 differentiable manifold M has already been constructed as a
 non-free module, which implies that the 2-dimensional
 differentiable manifold M is not parallelizable and hence cannot
 be the domain of a coordinate chart

Similarly, one cannot declare a vector frame on \(M\):

sage: e = M.vector_frame('e')
Traceback (most recent call last):
...
ValueError: the Module X(M) of vector fields on the 2-dimensional
 differentiable manifold M has already been constructed as a
 non-free module and therefore cannot have a basis

One shall use the keyword force_free=True to construct a free module before declaring the chart:

sage: M = Manifold(2, 'M')
sage: XM = M.vector_field_module(force_free=True)
sage: X.<x,y> = M.chart()  # OK
sage: e = M.vector_frame('e')  # OK

If one declares the chart or the vector frame before asking for the vector field module, the latter is initialized as a free module, without the need to specify force_free=True. Indeed, the information that \(M\) is the domain of a chart or a vector frame implies that \(M\) is parallelizable and is therefore sufficient to assert that \(\mathfrak{X}(M)\) is a free module over \(C^k(M)\):

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: XM = M.vector_field_module()
sage: isinstance(XM, FiniteRankFreeModule)
True
sage: M.is_manifestly_parallelizable()
True

vector_frame(*args, **kwargs)#

Define a vector frame on self.

A vector frame is a field on the manifold that provides, at each point \(p\) of the manifold, a vector basis of the tangent space at \(p\) (or at \(\Phi(p)\) when dest_map is not None, see below).

The vector frame can be defined from a set of \(n\) linearly independent vector fields, \(n\) being the dimension of self.

See also

VectorFrame for complete documentation.

INPUT:

symbol – either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual symbols of the vector fields; can be omitted only if from_frame is not None (see below)
vector_fields – tuple or list of \(n\) linearly independent vector fields on the manifold self (\(n\) being the dimension of self) defining the vector frame; can be omitted if the vector frame is created from scratch or if from_frame is not None
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual LaTeX symbols of the vector fields; if None, symbol is used in place of latex_symbol
dest_map – (default: None) DiffMap; destination map \(\Phi:\ U \rightarrow M\), where \(U\) is self and \(M\) is a differentiable manifold; for each \(p\in U\), the vector frame evaluated at \(p\) is a basis of the tangent space \(T_{\Phi(p)}M\); if dest_map is None, the identity map is assumed (case of a vector frame on \(U\))
from_frame – (default: None) vector frame \(\tilde{e}\) on the codomain \(M\) of the destination map \(\Phi\); the returned frame \(e\) is then such that for all \(p \in U\), we have \(e(p) = \tilde{e}(\Phi(p))\)
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the vector fields of the frame; if None, the indices will be generated as integers within the range declared on self
latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the vector fields; if None, indices is used instead
symbol_dual – (default: None) same as symbol but for the dual coframe; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual coframe
latex_symbol_dual – (default: None) same as latex_symbol but for the dual coframe

OUTPUT:

a VectorFrame representing the defined vector frame

EXAMPLES:

Defining a vector frame from two linearly independent vector fields on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: e0 = M.vector_field(1+x^2, 1+y^2)
sage: e1 = M.vector_field(2, -x*y)
sage: e = M.vector_frame('e', (e0, e1)); e
Vector frame (M, (e_0,e_1))
sage: e[0].display()
e_0 = (x^2 + 1) ∂/∂x + (y^2 + 1) ∂/∂y
sage: e[1].display()
e_1 = 2 ∂/∂x - x*y ∂/∂y
sage: (e[0], e[1]) == (e0, e1)
True

If the vector fields are not linearly independent, an error is raised:

sage: z = M.vector_frame('z', (e0, -e0))
Traceback (most recent call last):
...
ValueError: the provided vector fields are not linearly
 independent

Another example, involving a pair vector fields along a curve:

sage: R.<t> = manifolds.RealLine()
sage: c = M.curve([sin(t), sin(2*t)/2], (t, 0, 2*pi), name='c')
sage: I = c.domain(); I
Real interval (0, 2*pi)
sage: v = c.tangent_vector_field()
sage: v.display()
c' = cos(t) ∂/∂x + (2*cos(t)^2 - 1) ∂/∂y
sage: w = I.vector_field(1-2*cos(t)^2, cos(t), dest_map=c)
sage: u = I.vector_frame('u', (v, w))
sage: u[0].display()
u_0 = cos(t) ∂/∂x + (2*cos(t)^2 - 1) ∂/∂y
sage: u[1].display()
u_1 = (-2*cos(t)^2 + 1) ∂/∂x + cos(t) ∂/∂y
sage: (u[0], u[1]) == (v, w)
True

It is also possible to create a vector frame from scratch, without connecting it to previously defined vector frames or vector fields (this can still be performed later via the method set_change_of_frame()):

sage: f = M.vector_frame('f'); f
Vector frame (M, (f_0,f_1))
sage: f[0]
Vector field f_0 on the 2-dimensional differentiable manifold M

Thanks to the keywords dest_map and from_frame, one can also define a vector frame from one preexisting on another manifold, via a differentiable map (here provided by the curve c):

sage: fc = I.vector_frame(dest_map=c, from_frame=f); fc
Vector frame ((0, 2*pi), (f_0,f_1)) with values on the
 2-dimensional differentiable manifold M
sage: fc[0]
Vector field f_0 along the Real interval (0, 2*pi) with values on
 the 2-dimensional differentiable manifold M

Note that the symbol for fc, namely \(f\), is inherited from f, the original vector frame.

See also

For more options, in particular for the choice of symbols and indices, see VectorFrame.