Sets of Morphisms between Differentiable Manifolds¶

The class DifferentiableManifoldHomset implements sets of morphisms between two differentiable manifolds over the same topological field $$K$$ (in most applications, $$K = \RR$$ or $$K = \CC$$), a morphism being a differentiable map for the category of differentiable manifolds.

The subclass DifferentiableCurveSet is devoted to the specific case of differential curves, i.e. morphisms whose domain is an open interval of $$\RR$$.

The subclass IntegratedCurveSet is devoted to differentiable curves that are defined as a solution to a system of second order differential equations.

The subclass IntegratedAutoparallelCurveSet is devoted to differentiable curves that are defined as autoparallel curves with respect to a certain affine connection.

The subclass IntegratedGeodesicSet is devoted to differentiable curves that are defined as geodesics with respect to to a certain metric.

AUTHORS:

• Eric Gourgoulhon (2015): initial version
• Travis Scrimshaw (2016): review tweaks
• Karim Van Aelst (2017): sets of integrated curves

REFERENCES:

class sage.manifolds.differentiable.manifold_homset.DifferentiableCurveSet(domain, codomain, name=None, latex_name=None)

Set of differentiable curves in a differentiable manifold.

Given an open interval $$I$$ of $$\RR$$ (possibly $$I = \RR$$) and a differentiable manifold $$M$$ over $$\RR$$, this is the set $$\mathrm{Hom}(I,M)$$ of morphisms (i.e. differentiable curves) $$I \to M$$.

INPUT:

• domainOpenInterval if an open interval $$I \subset \RR$$ (domain of the morphisms), or RealLine if $$I = \RR$$
• codomainDifferentiableManifold; differentiable manifold $$M$$ (codomain of the morphisms)
• name – (default: None) string; name given to the set of curves; if None, Hom(I, M) will be used
• latex_name – (default: None) string; LaTeX symbol to denote the set of curves; if None, $$\mathrm{Hom}(I,M)$$ will be used

EXAMPLES:

Set of curves $$\RR \longrightarrow M$$, where $$M$$ is a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine() ; R
Real number line R
sage: H = Hom(R, M) ; H
Set of Morphisms from Real number line R to 2-dimensional
differentiable manifold M in Category of smooth manifolds over Real
Field with 53 bits of precision
sage: H.category()
Category of homsets of topological spaces
sage: latex(H)
\mathrm{Hom}\left(\Bold{R},M\right)
sage: H.domain()
Real number line R
sage: H.codomain()
2-dimensional differentiable manifold M

An element of H is a curve in M:

sage: c = H.an_element(); c
Curve in the 2-dimensional differentiable manifold M
sage: c.display()
R --> M
t |--> (x, y) = (1/(t^2 + 1) - 1/2, 0)

The test suite is passed:

sage: TestSuite(H).run()

The set of curves $$(0,1) \longrightarrow U$$, where $$U$$ is an open subset of $$M$$:

sage: I = R.open_interval(0, 1) ; I
Real interval (0, 1)
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
sage: H = Hom(I, U) ; H
Set of Morphisms from Real interval (0, 1) to Open subset U of the
2-dimensional differentiable manifold M in Join of Category of
subobjects of sets and Category of smooth manifolds over Real Field
with 53 bits of precision

An element of H is a curve in U:

sage: c = H.an_element() ; c
Curve in the Open subset U of the 2-dimensional differentiable
manifold M
sage: c.display()
(0, 1) --> U
t |--> (x, y) = (1/(t^2 + 1) - 1/2, 0)

The set of curves $$\RR \longrightarrow \RR$$ is a set of (manifold) endomorphisms:

sage: E = Hom(R, R) ; E
Set of Morphisms from Real number line R to Real number line R in
Category of smooth manifolds over Real Field with 53 bits of precision
sage: E.category()
Category of endsets of topological spaces
sage: E.is_endomorphism_set()
True
sage: E is End(R)
True

It is a monoid for the law of morphism composition:

sage: E in Monoids()
True

The identity element of the monoid is the identity map of $$\RR$$:

sage: E.one()
Identity map Id_R of the Real number line R
sage: E.one() is R.identity_map()
True
sage: E.one().display()
Id_R: R --> R
t |--> t

A “typical” element of the monoid:

sage: E.an_element().display()
R --> R
t |--> 1/(t^2 + 1) - 1/2

The test suite is passed by E:

sage: TestSuite(E).run()

Similarly, the set of curves $$I \longrightarrow I$$ is a monoid, whose elements are (manifold) endomorphisms:

sage: EI = Hom(I, I) ; EI
Set of Morphisms from Real interval (0, 1) to Real interval (0, 1) in
Join of Category of subobjects of sets and
Category of smooth manifolds over Real Field with 53 bits of precision
sage: EI.category()
Category of endsets of subobjects of sets and topological spaces
sage: EI is End(I)
True
sage: EI in Monoids()
True

The identity element and a “typical” element of this monoid:

sage: EI.one()
Identity map Id_(0, 1) of the Real interval (0, 1)
sage: EI.one().display()
Id_(0, 1): (0, 1) --> (0, 1)
t |--> t
sage: EI.an_element().display()
(0, 1) --> (0, 1)
t |--> 1/2/(t^2 + 1) + 1/4

The test suite is passed by EI:

sage: TestSuite(EI).run()
Element
class sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset(domain, codomain, name=None, latex_name=None)

Set of differentiable maps between two differentiable manifolds.

Given two differentiable manifolds $$M$$ and $$N$$ over a topological field $$K$$, the class DifferentiableManifoldHomset implements the set $$\mathrm{Hom}(M,N)$$ of morphisms (i.e. differentiable maps) $$M\rightarrow N$$.

This is a Sage parent class, whose element class is DiffMap.

INPUT:

• domain – differentiable manifold $$M$$ (domain of the morphisms), as an instance of DifferentiableManifold
• codomain – differentiable manifold $$N$$ (codomain of the morphisms), as an instance of DifferentiableManifold
• name – (default: None) string; name given to the homset; if None, Hom(M,N) will be used
• latex_name – (default: None) string; LaTeX symbol to denote the homset; if None, $$\mathrm{Hom}(M,N)$$ will be used

EXAMPLES:

Set of differentiable maps between a 2-dimensional differentiable manifold and a 3-dimensional one:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: H = Hom(M, N) ; H
Set of Morphisms from 2-dimensional differentiable manifold M to
3-dimensional differentiable manifold N in Category of smooth
manifolds over Real Field with 53 bits of precision
sage: type(H)
<class 'sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset_with_category'>
sage: H.category()
Category of homsets of topological spaces
sage: latex(H)
\mathrm{Hom}\left(M,N\right)
sage: H.domain()
2-dimensional differentiable manifold M
sage: H.codomain()
3-dimensional differentiable manifold N

An element of H is a differentiable map from M to N:

sage: H.Element
<class 'sage.manifolds.differentiable.diff_map.DiffMap'>
sage: f = H.an_element() ; f
Differentiable map from the 2-dimensional differentiable manifold M to the
3-dimensional differentiable manifold N
sage: f.display()
M --> N
(x, y) |--> (u, v, w) = (0, 0, 0)

The test suite is passed:

sage: TestSuite(H).run()

When the codomain coincides with the domain, the homset is a set of endomorphisms in the category of differentiable manifolds:

sage: E = Hom(M, M) ; E
Set of Morphisms from 2-dimensional differentiable manifold M to
2-dimensional differentiable manifold M in Category of smooth
manifolds over Real Field with 53 bits of precision
sage: E.category()
Category of endsets of topological spaces
sage: E.is_endomorphism_set()
True
sage: E is End(M)
True

In this case, the homset is a monoid for the law of morphism composition:

sage: E in Monoids()
True

This was of course not the case for H = Hom(M, N):

sage: H in Monoids()
False

The identity element of the monoid is of course the identity map of M:

sage: E.one()
Identity map Id_M of the 2-dimensional differentiable manifold M
sage: E.one() is M.identity_map()
True
sage: E.one().display()
Id_M: M --> M
(x, y) |--> (x, y)

The test suite is passed by E:

sage: TestSuite(E).run()

This test suite includes more tests than in the case of H, since E has some extra structure (monoid).

Element
class sage.manifolds.differentiable.manifold_homset.IntegratedAutoparallelCurveSet(domain, codomain, name=None, latex_name=None)

Set of integrated autoparallel curves in a differentiable manifold.

INPUT:

• domainOpenInterval open interval $$I \subset \RR$$ with finite boundaries (domain of the morphisms)
• codomainDifferentiableManifold; differentiable manifold $$M$$ (codomain of the morphisms)
• name – (default: None) string; name given to the set of integrated autoparallel curves; if None, Hom_autoparallel(I, M) will be used
• latex_name – (default: None) string; LaTeX symbol to denote the set of integrated autoparallel curves; if None, $$\mathrm{Hom_{autoparallel}}(I,M)$$ will be used

EXAMPLES:

This parent class needs to be imported:

sage: from sage.manifolds.differentiable.manifold_homset import IntegratedAutoparallelCurveSet

Integrated autoparallel curves are only allowed to be defined on an interval with finite bounds. This forbids to define an instance of this parent class whose domain has infinite bounds:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: H = IntegratedAutoparallelCurveSet(R, M)
Traceback (most recent call last):
...
ValueError: both boundaries of the interval defining the domain
of a Homset of integrated autoparallel curves need to be finite

An instance whose domain is an interval with finite bounds allows to build a curve that is autoparallel with respect to a connection defined on the codomain:

sage: I = R.open_interval(-1, 2)
sage: H = IntegratedAutoparallelCurveSet(I, M) ; H
Set of Morphisms from Real interval (-1, 2) to 2-dimensional
differentiable manifold M in Category of homsets of subobjects
of sets and topological spaces which actually are integrated
autoparallel curves with respect to a certain affine connection
sage: nab = M.affine_connection('nabla')
sage: nab[0,1,0], nab[0,0,1] = 1,2
sage: nab.torsion()[:]
[[[0, -1], [1, 0]], [[0, 0], [0, 0]]]
sage: t = var('t')
sage: p = M.point((3,4))
sage: Tp = M.tangent_space(p)
sage: v = Tp((1,2))
sage: c = H(nab, t, v, name='c') ; c
Integrated autoparallel curve c in the 2-dimensional
differentiable manifold M

A “typical” element of H is an autoparallel curve in M:

sage: d = H.an_element(); d
Integrated autoparallel curve in the 2-dimensional
differentiable manifold M
sage: sys = d.system(verbose=True)
Autoparallel curve in the 2-dimensional differentiable manifold
M equipped with Affine connection nab on the 2-dimensional
differentiable manifold M, and integrated over the Real
interval (-1, 2) as a solution to the following equations,
written with respect to Chart (M, (x, y)):

Initial point: Point on the 2-dimensional differentiable
manifold M with coordinates [0, -1/2] with respect to
Chart (M, (x, y))
Initial tangent vector: Tangent vector at Point on the
2-dimensional differentiable manifold M with components
[-1/6/(e^(-1) - 1), 1/3] with respect to Chart (M, (x, y))

d(x)/dt = Dx
d(y)/dt = Dy
d(Dx)/dt = -Dx*Dy
d(Dy)/dt = 0

The test suite is passed:

sage: TestSuite(H).run()

For any open interval $$J$$ with finite bounds $$(a,b)$$, all curves are autoparallel with respect to any connection. Therefore, the set of autoparallel curves $$J \longrightarrow J$$ is a set of numerical (manifold) endomorphisms that is a monoid for the law of morphism composition:

sage: [a,b] = var('a b')
sage: J = R.open_interval(a, b)
sage: H = IntegratedAutoparallelCurveSet(J, J); H
Set of Morphisms from Real interval (a, b) to Real interval
(a, b) in Category of endsets of subobjects of sets and
topological spaces which actually are integrated autoparallel
curves with respect to a certain affine connection
sage: H.category()
Category of endsets of subobjects of sets and topological spaces
sage: H in Monoids()
True

Although it is a monoid, no identity map is implemented via the one method of this class or its subclass devoted to geodesics. This is justified by the lack of relevance of the identity map within the framework of this parent class and its subclass, whose purpose is mainly devoted to numerical issues (therefore, the user is left free to set a numerical version of the identity if needed):

sage: H.one()
Traceback (most recent call last):
...
ValueError: the identity is not implemented for integrated
curves and associated subclasses

A “typical” element of the monoid:

sage: g = H.an_element() ; g
Integrated autoparallel curve in the Real interval (a, b)
sage: sys = g.system(verbose=True)
Autoparallel curve in the Real interval (a, b) equipped with
Affine connection nab on the Real interval (a, b), and
integrated over the Real interval (a, b) as a solution to the
following equations, written with respect to Chart ((a, b), (t,)):

Initial point: Point on the Real number line R with coordinates
 with respect to Chart ((a, b), (t,))
Initial tangent vector: Tangent vector at Point on the Real
number line R with components
[-(e^(1/2) - 1)/(a - b)] with respect to
Chart ((a, b), (t,))

d(t)/ds = Dt
d(Dt)/ds = -Dt^2

The test suite is passed, tests _test_one and _test_prod being skipped for reasons mentioned above:

sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
Element
class sage.manifolds.differentiable.manifold_homset.IntegratedCurveSet(domain, codomain, name=None, latex_name=None)

Set of integrated curves in a differentiable manifold.

INPUT:

• domainOpenInterval open interval $$I \subset \RR$$ with finite boundaries (domain of the morphisms)
• codomainDifferentiableManifold; differentiable manifold $$M$$ (codomain of the morphisms)
• name – (default: None) string; name given to the set of integrated curves; if None, Hom_integrated(I, M) will be used
• latex_name – (default: None) string; LaTeX symbol to denote the set of integrated curves; if None, $$\mathrm{Hom_{integrated}}(I,M)$$ will be used

EXAMPLES:

This parent class needs to be imported:

sage: from sage.manifolds.differentiable.manifold_homset import IntegratedCurveSet

Integrated curves are only allowed to be defined on an interval with finite bounds. This forbids to define an instance of this parent class whose domain has infinite bounds:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: H = IntegratedCurveSet(R, M)
Traceback (most recent call last):
...
ValueError: both boundaries of the interval defining the domain
of a Homset of integrated curves need to be finite

An instance whose domain is an interval with finite bounds allows to build an integrated curve defined on the interval:

sage: I = R.open_interval(-1, 2)
sage: H = IntegratedCurveSet(I, M) ; H
Set of Morphisms from Real interval (-1, 2) to 2-dimensional
differentiable manifold M in Category of homsets of subobjects
of sets and topological spaces which actually are integrated
curves
sage: eqns_rhs = [1,1]
sage: vels = X.symbolic_velocities()
sage: t = var('t')
sage: p = M.point((3,4))
sage: Tp = M.tangent_space(p)
sage: v = Tp((1,2))
sage: c = H(eqns_rhs, vels, t, v, name='c') ; c
Integrated curve c in the 2-dimensional differentiable
manifold M

A “typical” element of H is a curve in M:

sage: d = H.an_element(); d
Integrated curve in the 2-dimensional differentiable manifold M
sage: sys = d.system(verbose=True)
Curve in the 2-dimensional differentiable manifold M integrated
over the Real interval (-1, 2) as a solution to the following
system, written with respect to Chart (M, (x, y)):

Initial point: Point on the 2-dimensional differentiable
manifold M with coordinates [0, 0] with respect to Chart (M, (x, y))
Initial tangent vector: Tangent vector at Point on the
2-dimensional differentiable manifold M with components
[1/4, 0] with respect to Chart (M, (x, y))

d(x)/dt = Dx
d(y)/dt = Dy
d(Dx)/dt = -1/4*sin(t + 1)
d(Dy)/dt = 0

The test suite is passed:

sage: TestSuite(H).run()

More generally, an instance of this class may be defined with abstract bounds $$(a,b)$$:

sage: [a,b] = var('a b')
sage: J = R.open_interval(a, b)
sage: H = IntegratedCurveSet(J, M) ; H
Set of Morphisms from Real interval (a, b) to 2-dimensional
differentiable manifold M in Category of homsets of subobjects
of sets and topological spaces which actually are integrated
curves

A “typical” element of H is a curve in M:

sage: f = H.an_element(); f
Integrated curve in the 2-dimensional differentiable manifold M
sage: sys = f.system(verbose=True)
Curve in the 2-dimensional differentiable manifold M integrated
over the Real interval (a, b) as a solution to the following
system, written with respect to Chart (M, (x, y)):

Initial point: Point on the 2-dimensional differentiable
manifold M with coordinates [0, 0] with respect to Chart (M, (x, y))
Initial tangent vector: Tangent vector at Point on the
2-dimensional differentiable manifold M with components
[1/4, 0] with respect to Chart (M, (x, y))

d(x)/dt = Dx
d(y)/dt = Dy
d(Dx)/dt = -1/4*sin(-a + t)
d(Dy)/dt = 0

Yet, even in the case of abstract bounds, considering any of them to be infinite is still prohibited since no numerical integration could be performed:

sage: f.solve(parameters_values={a:-1, b:+oo})
Traceback (most recent call last):
...
ValueError: both boundaries of the interval need to be finite

The set of integrated curves $$J \longrightarrow J$$ is a set of numerical (manifold) endomorphisms:

sage: H = IntegratedCurveSet(J, J); H
Set of Morphisms from Real interval (a, b) to Real interval
(a, b) in Category of endsets of subobjects of sets and
topological spaces which actually are integrated curves
sage: H.category()
Category of endsets of subobjects of sets and topological spaces

It is a monoid for the law of morphism composition:

sage: H in Monoids()
True

Although it is a monoid, no identity map is implemented via the one method of this class or any of its subclasses. This is justified by the lack of relevance of the identity map within the framework of this parent class and its subclasses, whose purpose is mainly devoted to numerical issues (therefore, the user is left free to set a numerical version of the identity if needed):

sage: H.one()
Traceback (most recent call last):
...
ValueError: the identity is not implemented for integrated
curves and associated subclasses

A “typical” element of the monoid:

sage: g = H.an_element() ; g
Integrated curve in the Real interval (a, b)
sage: sys = g.system(verbose=True)
Curve in the Real interval (a, b) integrated over the Real
interval (a, b) as a solution to the following system, written
with respect to Chart ((a, b), (t,)):

Initial point: Point on the Real number line R with coordinates
 with respect to Chart ((a, b), (t,))
Initial tangent vector: Tangent vector at Point on the Real
number line R with components [1/4] with respect to
Chart ((a, b), (t,))

d(t)/ds = Dt
d(Dt)/ds = -1/4*sin(-a + s)

The test suite is passed, tests _test_one and _test_prod being skipped for reasons mentioned above:

sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
Element
one()

Raise an error refusing to provide the identity element. This overrides the one method of class TopologicalManifoldHomset, which would actually raise an error as well, due to lack of option is_identity in element_constructor method of self.

class sage.manifolds.differentiable.manifold_homset.IntegratedGeodesicSet(domain, codomain, name=None, latex_name=None)

Set of integrated geodesic in a differentiable manifold.

INPUT:

• domainOpenInterval open interval $$I \subset \RR$$ with finite boundaries (domain of the morphisms)
• codomainDifferentiableManifold; differentiable manifold $$M$$ (codomain of the morphisms)
• name – (default: None) string; name given to the set of integrated geodesics; if None, Hom_geodesic(I, M) will be used
• latex_name – (default: None) string; LaTeX symbol to denote the set of integrated geodesics; if None, $$\mathrm{Hom_{geodesic}}(I,M)$$ will be used

EXAMPLES:

This parent class needs to be imported:

sage: from sage.manifolds.differentiable.manifold_homset import IntegratedGeodesicSet

Integrated geodesics are only allowed to be defined on an interval with finite bounds. This forbids to define an instance of this parent class whose domain has infinite bounds:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: R.<t> = RealLine()
sage: H = IntegratedGeodesicSet(R, M)
Traceback (most recent call last):
...
ValueError: both boundaries of the interval defining the domain
of a Homset of integrated geodesics need to be finite

An instance whose domain is an interval with finite bounds allows to build a geodesic with respect to a metric defined on the codomain:

sage: I = R.open_interval(-1, 2)
sage: H = IntegratedGeodesicSet(I, M) ; H
Set of Morphisms from Real interval (-1, 2) to 2-dimensional
differentiable manifold M in Category of homsets of subobjects
of sets and topological spaces which actually are integrated
geodesics with respect to a certain metric
sage: g = M.metric('g')
sage: g[0,0], g[1,1], g[0,1] = 1, 1, 2
sage: t = var('t')
sage: p = M.point((3,4))
sage: Tp = M.tangent_space(p)
sage: v = Tp((1,2))
sage: c = H(g, t, v, name='c') ; c
Integrated geodesic c in the 2-dimensional differentiable
manifold M

A “typical” element of H is a geodesic in M:

sage: d = H.an_element(); d
Integrated geodesic in the 2-dimensional differentiable
manifold M
sage: sys = d.system(verbose=True)
Geodesic in the 2-dimensional differentiable manifold M equipped
with Riemannian metric g on the 2-dimensional differentiable
manifold M, and integrated over the Real interval (-1, 2) as a
solution to the following geodesic equations, written
with respect to Chart (M, (x, y)):

Initial point: Point on the 2-dimensional differentiable
manifold M with coordinates [0, 0] with respect to
Chart (M, (x, y))
Initial tangent vector: Tangent vector at Point on the
2-dimensional differentiable manifold M with components
[1/3*e^(1/2) - 1/3, 0] with respect to Chart (M, (x, y))

d(x)/dt = Dx
d(y)/dt = Dy
d(Dx)/dt = -Dx^2
d(Dy)/dt = 0

The test suite is passed:

sage: TestSuite(H).run()

For any open interval $$J$$ with finite bounds $$(a,b)$$, all curves are geodesics with respect to any metric. Therefore, the set of geodesics $$J \longrightarrow J$$ is a set of numerical (manifold) endomorphisms that is a monoid for the law of morphism composition:

sage: [a,b] = var('a b')
sage: J = R.open_interval(a, b)
sage: H = IntegratedGeodesicSet(J, J); H
Set of Morphisms from Real interval (a, b) to Real interval
(a, b) in Category of endsets of subobjects of sets and
topological spaces which actually are integrated geodesics
with respect to a certain metric
sage: H.category()
Category of endsets of subobjects of sets and topological spaces
sage: H in Monoids()
True

Although it is a monoid, no identity map is implemented via the one method of this class. This is justified by the lack of relevance of the identity map within the framework of this parent class, whose purpose is mainly devoted to numerical issues (therefore, the user is left free to set a numerical version of the identity if needed):

sage: H.one()
Traceback (most recent call last):
...
ValueError: the identity is not implemented for integrated
curves and associated subclasses

A “typical” element of the monoid:

sage: g = H.an_element() ; g
Integrated geodesic in the Real interval (a, b)
sage: sys = g.system(verbose=True)
Geodesic in the Real interval (a, b) equipped with Riemannian
metric g on the Real interval (a, b), and integrated over the
Real interval (a, b) as a solution to the following geodesic
equations, written with respect to Chart ((a, b), (t,)):

Initial point: Point on the Real number line R with coordinates
 with respect to Chart ((a, b), (t,))
Initial tangent vector: Tangent vector at Point on the Real
number line R with components [-(e^(1/2) - 1)/(a - b)]
with respect to Chart ((a, b), (t,))

d(t)/ds = Dt
d(Dt)/ds = -Dt^2

The test suite is passed, tests _test_one and _test_prod being skipped for reasons mentioned above:

sage: TestSuite(H).run(skip=["_test_one", "_test_prod"])
Element