Manifold Structures¶
These classes encode the structure of a manifold.
AUTHORS:
Travis Scrimshaw (2015-11-25): Initial version
Eric Gourgoulhon (2015): add
DifferentialStructure
andRealDifferentialStructure
Eric Gourgoulhon (2018): add
PseudoRiemannianStructure
,RiemannianStructure
andLorentzianStructure
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class
sage.manifolds.structure.
DegenerateStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a degenerate manifold.
-
chart
¶
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scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
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subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import DegenerateStructure sage: from sage.categories.manifolds import Manifolds sage: DegenerateStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
DifferentialStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a differentiable manifold over a general topological field.
-
chart
¶
-
scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import DifferentialStructure sage: from sage.categories.manifolds import Manifolds sage: DifferentialStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
LorentzianStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a Lorentzian manifold.
-
chart
¶
-
scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import LorentzianStructure sage: from sage.categories.manifolds import Manifolds sage: LorentzianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
PseudoRiemannianStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a pseudo-Riemannian manifold.
-
chart
¶
-
scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import PseudoRiemannianStructure sage: from sage.categories.manifolds import Manifolds sage: PseudoRiemannianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
RealDifferentialStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a differentiable manifold over \(\RR\).
-
chart
¶
-
scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RealDifferentialStructure sage: from sage.categories.manifolds import Manifolds sage: RealDifferentialStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
RealTopologicalStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a topological manifold over \(\RR\).
-
chart
¶ alias of
sage.manifolds.chart.RealChart
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homset
¶ alias of
sage.manifolds.manifold_homset.TopologicalManifoldHomset
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scalar_field_algebra
¶ alias of
sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RealTopologicalStructure sage: from sage.categories.manifolds import Manifolds sage: RealTopologicalStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
RiemannianStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a Riemannian manifold.
-
chart
¶
-
scalar_field_algebra
¶ alias of
sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RiemannianStructure sage: from sage.categories.manifolds import Manifolds sage: RiemannianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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-
class
sage.manifolds.structure.
TopologicalStructure
¶ Bases:
sage.misc.fast_methods.Singleton
The structure of a topological manifold over a general topological field.
-
chart
¶ alias of
sage.manifolds.chart.Chart
-
homset
¶ alias of
sage.manifolds.manifold_homset.TopologicalManifoldHomset
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scalar_field_algebra
¶ alias of
sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra
-
subcategory
(cat)¶ Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import TopologicalStructure sage: from sage.categories.manifolds import Manifolds sage: TopologicalStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
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