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Manifolds
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Sage 9.8 Reference Manual
  • Home - Manifolds
  • Topological Manifolds
    • Topological Manifolds
    • Subsets of Topological Manifolds
    • Manifold Structures
    • Points of Topological Manifolds
    • Coordinate Charts
      • Coordinate Charts
      • Chart Functions
      • Coordinate calculus methods
    • Scalar Fields
      • Algebra of Scalar Fields
      • Scalar Fields
    • Continuous Maps
      • Sets of Morphisms between Topological Manifolds
      • Continuous Maps Between Topological Manifolds
      • Images of Manifold Subsets under Continuous Maps as Subsets of the Codomain
    • Submanifolds of topological manifolds
    • Topological Vector Bundles
      • Topological Vector Bundle
      • Vector Bundle Fibers
      • Vector Bundle Fiber Elements
      • Trivializations
      • Local Frames
      • Section Modules
      • Sections
    • Families of Manifold Objects
    • Topological Closures of Manifold Subsets
    • Manifold Subsets Defined as Pullbacks of Subsets under Continuous Maps
  • Differentiable Manifolds
    • Differentiable Manifolds
    • Coordinate Charts on Differentiable Manifolds
    • The Real Line and Open Intervals
    • Scalar Fields
      • Algebra of Differentiable Scalar Fields
      • Differentiable Scalar Fields
    • Differentiable Maps and Curves
      • Sets of Morphisms between Differentiable Manifolds
      • Differentiable Maps between Differentiable Manifolds
      • Curves in Manifolds
      • Integrated Curves and Geodesics in Manifolds
    • Tangent Spaces
      • Tangent Spaces
      • Tangent Vectors
    • Vector Fields
      • Vector Field Modules
      • Vector Fields
      • Vector Frames
      • Group of Tangent-Space Automorphism Fields
      • Tangent-Space Automorphism Fields
    • Tensor Fields
      • Tensor Field Modules
      • Tensor Fields
      • Tensor Fields with Values on a Parallelizable Manifold
    • Differential Forms
      • Differential Form Modules
      • Differential Forms
    • Mixed Differential Forms
      • Graded Algebra of Mixed Differential Forms
      • Mixed Differential Forms
    • De Rham Cohomology
    • Alternating Multivector Fields
      • Multivector Field Modules
      • Multivector Fields
    • Affine Connections
    • Submanifolds of differentiable manifolds
    • Differentiable Vector Bundles
      • Differentiable Vector Bundles
      • Bundle Connections
      • Characteristic cohomology classes
  • Pseudo-Riemannian Manifolds
    • Pseudo-Riemannian Manifolds
    • Euclidean Spaces
      • Euclidean Spaces
      • Spheres smoothly embedded in Euclidean Space
      • Operators for vector calculus
    • Pseudo-Riemannian Metrics and Degenerate Metrics
    • Levi-Civita Connections
    • Pseudo-Riemannian submanifolds
    • Degenerate Metric Manifolds
      • Degenerate manifolds
      • Degenerate submanifolds
  • Poisson Manifolds
    • Poisson tensors
    • Symplectic structures
    • Symplectic vector spaces
  • Utilities for Calculus
  • Manifolds Catalog
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Poisson Manifolds#

  • Poisson tensors
    • PoissonTensorField
      • hamiltonian_vector_field()
      • poisson_bracket()
      • sharp()
    • PoissonTensorFieldParal
  • Symplectic structures
    • SymplecticForm
      • flat()
      • hamiltonian_vector_field()
      • hodge_star()
      • poisson()
      • poisson_bracket()
      • restrict()
      • sharp()
      • volume_form()
      • wrap()
    • SymplecticFormParal
      • poisson()
      • restrict()
  • Symplectic vector spaces
    • StandardSymplecticSpace
      • symplectic_form()
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Poisson tensors
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Degenerate submanifolds
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