definition 13.48 Differentiable manifold

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2518 · p. 483

ground object -- no derivation owed

Rests on

depends_on definition 13.47 Differentiable structure ¶
- depends_on definition 13.44 Atlas ¶
  - depends_on definition 13.43 Coordinate system ¶

Supports

depends_on definition A.67 Symplectic manifold ¶
- depends_on theorem A.68 Darboux ¶
  - depends_on corollary A.69 No local invariants ¶
    - depends_on remark 24.13 What Darboux's theorem forbids ¶
  - depends_on example A.80 The phase space of one particle in space ¶
  - depends_on theorem 24.12 Darboux ¶
    - depends_on definition 24.20 Liouville volume ¶
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 24.21 Liouville ¶
      - depends_on theorem 24.24 Poincaré recurrence ¶
    - depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
depends_on definition A.299 Manifold with boundary ¶
- depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
  - depends_on definition A.309 The induced orientation of the boundary ¶
    - depends_on theorem A.311 General Stokes theorem ¶
      - depends_on corollary A.313 Stokes' theorem for an antisymmetric tensor field ¶
      - depends_on lemma A.555 The actions are well defined ¶
      - depends_on theorem 24.23 Poincaré–Cartan integral invariant ¶
- depends_on lemma A.307 Partition of unity on a compact manifold ¶
  - depends_on definition A.308 Integral over the manifold ¶
    - depends_on theorem A.311 General Stokes theorem ¶ ↺
depends_on definition 13.71 Differentiable curve ¶
- depends_on definition 13.124 Integral curve; complete vector field ¶
  - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
    - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
- depends_on definition 13.81 Vector on a manifold ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition A.70 Pullback ¶
      - depends_on definition 24.17 Symplectomorphism ¶
      - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma A.71 The pullback is an algebra map commuting with $\dd$ ¶
    - depends_on definition 13.98 $k$-form ¶
      - depends_on definition A.70 Pullback ¶ ↺
      - depends_on definition A.67 Symplectic manifold ¶ ↺
      - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.106 Exact form ¶
      - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.116 Hodge dual ¶
      - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
      - depends_on proposition 13.100 The space of $k$-forms ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 13.99 $k$-vector ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      - depends_on proposition 13.100 The space of $k$-forms ¶ ↺
    - depends_on proposition 13.100 The space of $k$-forms ¶ ↺
  - depends_on definition 13.52 Differential; pushforward ¶
    - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.54 Embedded submanifold ¶
      - depends_on example 13.55 An injective immersion that is not an embedding ¶
      - depends_on lemma A.533 The orbit map has constant rank $d$ ¶
      - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
      - depends_on theorem A.537 The smooth structure on the orbit space ¶
    - depends_on example 13.55 An injective immersion that is not an embedding ¶ ↺
    - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
      - depends_on remark 24.11 The SI dimension of every object in this chapter ¶
    - depends_on theorem 13.59 Regular value theorem ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
  - depends_on definition 13.99 $k$-vector ¶ ↺
  - depends_on definition 13.74 Length of a curve ¶
    - depends_on definition 13.76 Arclength parametrisation ¶
      - depends_on definition 13.77 Curvature vector ¶
    - depends_on proposition 13.75 Invariance of the length ¶
  - depends_on definition 13.84 Tensor ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 32.3 Dynamical system, phase space, flow ¶
      - depends_on definition 32.8 Attractor and basin ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on definition 32.10 Fixed point ¶
      - depends_on definition 32.41 The horseshoe map ¶
      - depends_on definition 32.83 The logistic map ¶
      - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on proposition 32.4 The flow is a one-parameter group ¶
      - depends_on theorem 32.6 Evolution of phase volume ¶
    - depends_on definition 13.145 Affine connection ¶
      - depends_on definition A.625 Four-dimensional extrinsic curvature ¶
      - depends_on definition A.624 Projector and induced metric ¶
      - depends_on definition 13.77 Curvature vector ¶ ↺
      - depends_on definition 13.149 Metric compatibility ¶
      - depends_on definition 13.146 Parallel transport and autoparallels ¶
      - depends_on definition 13.147 Torsion ¶
      - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
    - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on theorem 13.133 Frobenius ¶
    - depends_on definition 13.124 Integral curve; complete vector field ¶ ↺
    - depends_on example 13.86 A curve ¶
    - depends_on example 13.85 Euclidean space $E_{3}$ ¶
    - depends_on example 13.87 The $2$-sphere ¶
  - depends_on example 13.87 The $2$-sphere ¶ ↺
- depends_on example 13.86 A curve ¶ ↺
depends_on definition 13.66 Smooth action; free; proper; orbit ¶
- depends_on example 13.68 Why each hypothesis is there ¶
  - depends_on remark A.542 Where each hypothesis is spent ¶
- depends_on lemma A.531 The projection is open ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶
    - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
  - depends_on proposition A.532 Separation and countability of the quotient ¶
    - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
    - depends_on theorem A.537 The smooth structure on the orbit space ¶ ↺
- depends_on lemma A.533 The orbit map has constant rank $d$ ¶ ↺
- depends_on proposition A.535 Existence of a slice ¶
  - depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
  - depends_on remark A.542 Where each hypothesis is spent ¶ ↺
- depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
- depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on example 13.68 Why each hypothesis is there ¶ ↺
  - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
    - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
  - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
    - depends_on example A.572 Rotational reduction of the central-force problem ¶
depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
depends_on definition 13.49 Smooth map between manifolds ¶
- depends_on definition 13.50 Diffeomorphism ¶
- depends_on definition 13.52 Differential; pushforward ¶ ↺
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
- depends_on lemma A.538 Submersions have smooth local sections ¶ ↺
- depends_on proposition A.539 Universal property ¶
  - depends_on corollary A.540 Uniqueness of the smooth structure ¶
- depends_on theorem 13.59 Regular value theorem ¶ ↺
depends_on definition 13.54 Embedded submanifold ¶ ↺
depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on proposition 13.56 Product manifold ¶
- depends_on example 13.57 The cylinder ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiable structure	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2523
`depends_on`	←	Symplectic manifold	declared	appendices/A-long-proofs.tex:4422
`depends_on`	←	Manifold with boundary	declared	appendices/A-long-proofs.tex:14908
`depends_on`	←	Differentiable curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3351
`depends_on`	←	Smooth action; free; proper; orbit	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3062
`depends_on`	←	Metric tensor of signature $(p,q)$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5495
`depends_on`	←	Smooth map between manifolds	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2557
`depends_on`	←	Embedded submanifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2643
`depends_on`	←	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3916
`depends_on`	←	Product manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2711