Manifold Structures - Manifolds

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# Manifold Structures

These classes encode the structure of a manifold.

AUTHORS:

- Travis Scrimshaw (2015-11-25): Initial version

- Eric Gourgoulhon (2015): add `DifferentialStructure` and `RealDifferentialStructure`

- Eric Gourgoulhon (2018): add `PseudoRiemannianStructure`, `RiemannianStructure` and `LorentzianStructure`

class sage.manifolds.structure.DegenerateStructure

  Bases: `Singleton`

  The structure of a degenerate manifold.

  chart

    alias of `RealDiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'degenerate_metric'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import DegenerateStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> DegenerateStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.DifferentialStructure

  Bases: `Singleton`

  The structure of a differentiable manifold over a general topological field.

  chart

    alias of `DiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'differentiable'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import DifferentialStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> DifferentialStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.LorentzianStructure

  Bases: `Singleton`

  The structure of a Lorentzian manifold.

  chart

    alias of `RealDiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'Lorentzian'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import LorentzianStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> LorentzianStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.PseudoRiemannianStructure

  Bases: `Singleton`

  The structure of a pseudo-Riemannian manifold.

  chart

    alias of `RealDiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'pseudo-Riemannian'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import PseudoRiemannianStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> PseudoRiemannianStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.RealDifferentialStructure

  Bases: `Singleton`

  The structure of a differentiable manifold over \(\RR\).

  chart

    alias of `RealDiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'differentiable'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import RealDifferentialStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> RealDifferentialStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.RealTopologicalStructure

  Bases: `Singleton`

  The structure of a topological manifold over \(\RR\).

  chart

    alias of `RealChart`

  homset

    alias of `TopologicalManifoldHomset`

  name = 'topological'
  scalar_field_algebra

    alias of `ScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import RealTopologicalStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> RealTopologicalStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.RiemannianStructure

  Bases: `Singleton`

  The structure of a Riemannian manifold.

  chart

    alias of `RealDiffChart`

  homset

    alias of `DifferentiableManifoldHomset`

  name = 'Riemannian'
  scalar_field_algebra

    alias of `DiffScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import RiemannianStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> RiemannianStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision

class sage.manifolds.structure.TopologicalStructure

  Bases: `Singleton`

  The structure of a topological manifold over a general topological field.

  chart

    alias of `Chart`

  homset

    alias of `TopologicalManifoldHomset`

  name = 'topological'
  scalar_field_algebra

    alias of `ScalarFieldAlgebra`

  subcategory(cat)

    Return the subcategory of `cat` corresponding to the structure of `self`.

    EXAMPLES:

    Sage

        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

    Python

        >>> from sage.all import *
        >>> from sage.manifolds.structure import TopologicalStructure
        >>> from sage.categories.manifolds import Manifolds
        >>> TopologicalStructure().subcategory(Manifolds(RR))
        Category of manifolds over Real Field with 53 bits of precision