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 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)#
Bases:
DifferentiableManifoldHomset
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:
domain
–OpenInterval
if an open interval \(I \subset \RR\) (domain of the morphisms), orRealLine
if \(I = \RR\)codomain
–DifferentiableManifold
; differentiable manifold \(M\) (codomain of the morphisms)name
– (default:None
) string; name given to the set of curves; ifNone
,Hom(I, M)
will be usedlatex_name
– (default:None
) string; LaTeX symbol to denote the set of curves; ifNone
, \(\mathrm{Hom}(I,M)\) 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> = manifolds.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 inM
: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 inU
: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 connected 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 and Category of connected 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#
alias of
DifferentiableCurve
- class sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset(domain, codomain, name=None, latex_name=None)#
Bases:
TopologicalManifoldHomset
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 ofDifferentiableManifold
codomain
– differentiable manifold \(N\) (codomain of the morphisms), as an instance ofDifferentiableManifold
name
– (default:None
) string; name given to the homset; ifNone
, Hom(M,N) will be usedlatex_name
– (default:None
) string; LaTeX symbol to denote the homset; ifNone
, \(\mathrm{Hom}(M,N)\) 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 fromM
toN
: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
, sinceE
has some extra structure (monoid).
- class sage.manifolds.differentiable.manifold_homset.IntegratedAutoparallelCurveSet(domain, codomain, name=None, latex_name=None)#
Bases:
IntegratedCurveSet
Set of integrated autoparallel curves in a differentiable manifold.
INPUT:
domain
–OpenInterval
open interval \(I \subset \RR\) with finite boundaries (domain of the morphisms)codomain
–DifferentiableManifold
; differentiable manifold \(M\) (codomain of the morphisms)name
– (default:None
) string; name given to the set of integrated autoparallel curves; ifNone
,Hom_autoparallel(I, M)
will be usedlatex_name
– (default:None
) string; LaTeX symbol to denote the set of integrated autoparallel curves; ifNone
, \(\mathrm{Hom_{autoparallel}}(I,M)\) 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> = manifolds.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 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 inM
: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 [0] 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#
alias of
IntegratedAutoparallelCurve
- class sage.manifolds.differentiable.manifold_homset.IntegratedCurveSet(domain, codomain, name=None, latex_name=None)#
Bases:
DifferentiableCurveSet
Set of integrated curves in a differentiable manifold.
INPUT:
domain
–OpenInterval
open interval \(I \subset \RR\) with finite boundaries (domain of the morphisms)codomain
–DifferentiableManifold
; differentiable manifold \(M\) (codomain of the morphisms)name
– (default:None
) string; name given to the set of integrated curves; ifNone
,Hom_integrated(I, M)
will be usedlatex_name
– (default:None
) string; LaTeX symbol to denote the set of integrated curves; ifNone
, \(\mathrm{Hom_{integrated}}(I,M)\) 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> = manifolds.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 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 inM
: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 topological spaces which actually are integrated curves
A “typical” element of
H
is a curve inM
: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 [0] 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#
alias of
IntegratedCurve
- one()#
Raise an error refusing to provide the identity element. This overrides the
one
method of classTopologicalManifoldHomset
, which would actually raise an error as well, due to lack of optionis_identity
inelement_constructor
method ofself
.
- class sage.manifolds.differentiable.manifold_homset.IntegratedGeodesicSet(domain, codomain, name=None, latex_name=None)#
Bases:
IntegratedAutoparallelCurveSet
Set of integrated geodesic in a differentiable manifold.
INPUT:
domain
–OpenInterval
open interval \(I \subset \RR\) with finite boundaries (domain of the morphisms)codomain
–DifferentiableManifold
; differentiable manifold \(M\) (codomain of the morphisms)name
– (default:None
) string; name given to the set of integrated geodesics; ifNone
,Hom_geodesic(I, M)
will be usedlatex_name
– (default:None
) string; LaTeX symbol to denote the set of integrated geodesics; ifNone
, \(\mathrm{Hom_{geodesic}}(I,M)\) 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> = manifolds.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 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 inM
: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 [0] 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#
alias of
IntegratedGeodesic