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 \(K\).
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).
INPUT:
n
– positive integer; dimension of the manifoldname
– string; name (symbol) given to the manifoldfield
– field \(K\) on which the manifold is defined; allowed values are'real'
or an object of typeRealField
(e.g.,RR
) for a manifold over \(\RR\)'complex'
or an object of typeComplexField
(e.g.,CC
) for a manifold over \(\CC\)an object in the category of topological fields (see
Fields
andTopologicalSpaces
) for other types of manifolds
structure
– manifold structure (seeDifferentialStructure
orRealDifferentialStructure
)base_manifold
– (default:None
) if notNone
, must be a differentiable manifold; the created object is then an open subset ofbase_manifold
diff_degree
– (default:infinity
) degree \(k\) of differentiabilitylatex_name
– (default:None
) string; LaTeX symbol to denote the manifold; if none is provided, it is set toname
start_index
– (default: 0) integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g. coordinates in a chartcategory
– (default:None
) to specify the category; ifNone
,Manifolds(field).Differentiable()
(orManifolds(field).Smooth()
ifdiff_degree
=infinity
) is assumed (see the categoryManifolds
)unique_tag
– (default:None
) tag used to force the construction of a new object when all the other arguments have been used previously (withoutunique_tag
, theUniqueRepresentation
behavior inherited fromManifoldSubset
, viaTopologicalManifold
, 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 connectionlatex_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
andAutomorphismFieldParal
for a complete documentation.INPUT:
comp
– (optional) either the components of the field of automorphisms with respect to the vector frame specified by the argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the fieldlatex_name
– (default:None
) LaTeX symbol to denote the field; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field of automorphisms on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
AutomorphismField
(or if \(N\) is parallelizable, aAutomorphismFieldParal
) 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
andAutomorphismFieldParal
.
- 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 onself
.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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map, otherwisedest_map
must be aDiffMap
OUTPUT:
a
AutomorphismFieldParalGroup
(if \(N\) is parallelizable) or aAutomorphismFieldGroup
(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
andAutomorphismFieldGroup
.
- change_of_frame(frame1, frame2)#
Return a change of vector frames defined on
self
.INPUT:
frame1
– vector frame 1frame2
– 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; ifNone
, 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)
, wheret
is the curve parameter used incoord_expression
;t_min
is its minimal value;t_max
its maximal value;
if
t_min=-Infinity
andt_max=+Infinity
, they can be omitted andt
can be passed forparam
instead of the tuple(t, t_min, t_max)
chart
– (default:None
) chart on the manifold used for case (ii) above; ifNone
the default chart of the manifold is assumedname
– (default:None
) string; symbol given to the curvelatex_name
– (default:None
) string; LaTeX symbol to denote the curve; if none is provided,name
will be used
OUTPUT:
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
andt_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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of mixed forms on \(M\)), otherwisedest_map
must be aDiffMap
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 metriclatex_name
– (default:None
) LaTeX symbol to denote the metric; ifNone
, it is formed fromname
dest_map
– (default:None
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, 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
andDiffFormParal
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 argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the differential formlatex_name
– (default:None
) LaTeX symbol to denote the differential form; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a differential form on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
the \(p\)-form as a
DiffForm
(or if \(N\) is parallelizable, aDiffFormParal
)
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
andDiffFormParal
.
- 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 onself
.See also
DiffFormModule
for complete documentation.INPUT:
degree
– positive integer; the degree \(p\) of the differential formsdest_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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of differential forms on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
DiffFormModule
(or if \(N\) is parallelizable, aDiffFormFreeModule
) 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 notNone
, 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
andchart2
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 incoord_functions
for case (ii); if none is provided, the coordinates are assumed to refer to the manifold’s default chartchart2
– (default:None
; used only in case (ii) above) chart oncodomain
defining the arrival coordinates involved incoord_functions
for case (ii); if none is provided, the coordinates are assumed to refer to the default chart ofcodomain
name
– (default:None
) name given to the differentiable maplatex_name
– (default:None
) LaTeX symbol to denote the differentiable map; if none is provided, the LaTeX symbol is set toname
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). IfNone
, the current manifold is taken.coord_functions
– (default:None
) if notNone
, 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
andchart2
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 incoord_functions
for case (ii); if none is provided, the coordinates are assumed to refer to the manifold’s default chartchart2
– (default:None
; used only in case (ii) above) chart oncodomain
defining the arrival coordinates involved incoord_functions
for case (ii); if none is provided, the coordinates are assumed to refer to the default chart ofcodomain
name
– (default:None
) name given to the diffeomorphismlatex_name
– (default:None
) LaTeX symbol to denote the diffeomorphism; if none is provided, the LaTeX symbol is set toname
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 autoparallelcurve_param
– a tuple of the type(t, t_min, t_max)
, wheret
is the symbolic variable to be used as the parameter of the curve (the equations defining an instance ofIntegratedAutoparallelCurve
are such thatt
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 curvechart
– (default:None
) chart on the manifold in which the equations are given ; ifNone
the default chart of the manifold is assumedname
– (default:None
) string; symbol given to the curvelatex_name
– (default:None
) string; LaTeX symbol to denote the curve; if none is provided,name
will be used
OUTPUT:
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 onlyvelocities
– list of the symbolic expressions used inequations_rhs
to denote the velocitiescurve_param
– a tuple of the type(t, t_min, t_max)
, wheret
is the symbolic variable used inequations_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 curvechart
– (default:None
) chart on the manifold in which the equations are given; ifNone
the default chart of the manifold is assumedname
– (default:None
) string; symbol given to the curvelatex_name
– (default:None
) string; LaTeX symbol to denote the curve; if none is provided,name
will be used
OUTPUT:
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 geodesiccurve_param
– a tuple of the type(t, t_min, t_max)
, wheret
is the symbolic variable to be used as the parameter of the curve (the equations defining an instance ofIntegratedGeodesic
are such thatt
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 curvechart
– (default:None
) chart on the manifold in which the equations are given; ifNone
the default chart of the manifold is assumedname
– (default:None
) string; symbol given to the curvelatex_name
– (default:None
) string; LaTeX symbol to denote the curve; if none is provided,name
will be used
OUTPUT:
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
ifself
is known to be a parallelizable andFalse
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 metricsignature
– (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; ifNone
, it is formed fromname
dest_map
– (default:None
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, 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 metricsignature
– (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; ifsignature
is not provided, \(S\) is set to the manifold’s dimension (Riemannian signature)latex_name
– (default:None
) LaTeX symbol to denote the metric; ifNone
, it is formed fromname
dest_map
– (default:None
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, 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 formsname
– (default:None
) name given to the differential formlatex_name
– (default:None
) LaTeX symbol to denote the differential form; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a differential form on \(M\)), otherwisedest_map
must be aDiffMap
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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of mixed forms on \(M\)), otherwisedest_map
must be aDiffMap
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
andMultivectorFieldParal
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 argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the multivector fieldlatex_name
– (default:None
) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a multivector field on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
the \(p\)-vector field as a
MultivectorField
(or if \(N\) is parallelizable, aMultivectorFieldParal
)
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
andMultivectorFieldParal
.
- 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 onself
.See also
MultivectorModule
for complete documentation.INPUT:
degree
– positive integer; the degree \(p\) of the multivector fieldsdest_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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of multivector fields on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
MultivectorModule
(or if \(N\) is parallelizable, aMultivectorFreeModule
) 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
andDiffFormParal
for a complete documentation.INPUT:
comp
– (optional) either the components of 1-form with respect to the vector frame specified by the argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the 1-formlatex_name
– (default:None
) LaTeX symbol to denote the 1-form; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a 1-form on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
the 1-form as a
DiffForm
(or if \(N\) is parallelizable, aDiffFormParal
)
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
andDiffFormParal
.
- 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 subsetlatex_name
– (default:None
) LaTeX symbol to denote the subset; if none is provided, it is set toname
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: onlyself
) 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 asM
: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 ofX
toU
: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 inU
: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 metriclatex_name
– (default:None
) LaTeX symbol to denote the metric; ifNone
, it is formed fromname
dest_map
– (default:None
) instance of classDiffMap
representing the destination map \(\Phi:\ U \rightarrow M\), where \(U\) is the current manifold; ifNone
, 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 \((e_i)\) belowframe2
– frame 2, denoted \((f_i)\) belowchange_of_frame
– instance of classAutomorphismFieldParal
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 dictionaryself._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 ofself
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, coveringself
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 argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the fieldlatex_name
– (default:None
) LaTeX symbol to denote the field; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field on \(M\)), otherwisedest_map
must be an instance of instance of classDiffMap
OUTPUT:
a
TensorField
(or if \(N\) is parallelizable, aTensorFieldParal
) 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; ifNone
, 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 mapslatex_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}’ ifname
is ‘Id’ and toname
otherwisedest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a field of identity maps on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
AutomorphismField
(or if \(N\) is parallelizable, aAutomorphismFieldParal
) 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 manifoldbase_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 \(p\) onself
comp
– components of the vector with respect to the basis specified by the argumentbasis
, either as an iterable or as a sequence of \(n\) components, \(n\) being the dimension ofself
(see examples below)basis
– (default:None
)FreeModuleBasis
; basis of the tangent space at \(p\) with respect to which the components are defined; ifNone
, the default basis of the tangent space is usedname
– (default:None
) string; symbol given to the vectorlatex_name
– (default:None
) string; LaTeX symbol to denote the vector; ifNone
,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
isvector
: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
andlatex_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 bundlel
– the covariant rank of the tensor bundledest_map
– (default:None
) destination map \(\Phi:\ M \rightarrow N\) (type:DiffMap
) from which the tensor bundle is pulled back; ifNone
, 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-\((k,l)\) overself
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
andTensorFieldParal
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 argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the tensor fieldlatex_name
– (default:None
) LaTeX symbol to denote the tensor field; ifNone
, the LaTeX symbol is set toname
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 conventionposition=0
for the first argument; for instance:sym = (0,1)
for a symmetry between the 1st and 2nd argumentssym = [(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 forsym
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a tensor field on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
TensorField
(or if \(N\) is parallelizable, aTensorFieldParal
) 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
andTensorFieldParal
.
- 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 onself
.See also
TensorFieldModule
for a complete documentation.INPUT:
tensor_type
– pair \((k,l)\) with \(k\) being the contravariant rank and \(l\) the covariant rankdest_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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of tensor fields on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
TensorFieldModule
(or if \(N\) is parallelizable, aTensorFieldFreeModule
) 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 \(p\) onself
comp
– components of the vector with respect to the basis specified by the argumentbasis
, either as an iterable or as a sequence of \(n\) components, \(n\) being the dimension ofself
(see examples below)basis
– (default:None
)FreeModuleBasis
; basis of the tangent space at \(p\) with respect to which the components are defined; ifNone
, the default basis of the tangent space is usedname
– (default:None
) string; symbol given to the vectorlatex_name
– (default:None
) string; LaTeX symbol to denote the vector; ifNone
,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
isvector
: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
andlatex_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 bundlename
– name given to the total spacefield
– (default:'real'
) topological field giving the vector space structure to the fiberslatex_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
andVectorFieldParal
for a complete documentation.INPUT:
comp
– (optional) either the components of the vector field with respect to the vector frame specified by the argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumedname
– (default:None
) name given to the vector fieldlatex_name
– (default:None
) LaTeX symbol to denote the vector field; if none is provided, the LaTeX symbol is set toname
dest_map
– (default:None
) the destination map \(\Phi:\ M \rightarrow N\); ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of a vector field on \(M\)), otherwisedest_map
must be aDiffMap
OUTPUT:
a
VectorField
(or if \(N\) is parallelizable, aVectorFieldParal
) 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
andVectorFieldParal
.
- 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; ifNone
, it is assumed that \(N = M\) and that \(\Phi\) is the identity map (case of vector fields on \(M\)), otherwisedest_map
must be aDiffMap
force_free
– (default:False
) if set toTrue
, force the construction of a free module (this implies that \(N\) is parallelizable)
OUTPUT:
a
VectorFieldModule
(or if \(N\) is parallelizable, aVectorFieldFreeModule
) 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 notNone
, 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 iffrom_frame
is notNone
(see below)vector_fields
– tuple or list of \(n\) linearly independent vector fields on the manifoldself
(\(n\) being the dimension ofself
) defining the vector frame; can be omitted if the vector frame is created from scratch or iffrom_frame
is notNone
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; ifNone
,symbol
is used in place oflatex_symbol
dest_map
– (default:None
)DiffMap
; destination map \(\Phi:\ U \rightarrow M\), where \(U\) isself
and \(M\) is a differentiable manifold; for each \(p\in U\), the vector frame evaluated at \(p\) is a basis of the tangent space \(T_{\Phi(p)}M\); ifdest_map
isNone
, 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 ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual coframe; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual coframelatex_symbol_dual
– (default:None
) same aslatex_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
andfrom_frame
, one can also define a vector frame from one preexisting on another manifold, via a differentiable map (here provided by the curvec
):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 fromf
, the original vector frame.See also
For more options, in particular for the choice of symbols and indices, see
VectorFrame
.