definition 13.47 Differentiable structure

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2505 · p. 482

ground object -- no derivation owed

Rests on

depends_on definition 13.44 Atlas ¶
- depends_on definition 13.43 Coordinate system ¶

Supports

depends_on corollary A.540 Uniqueness of the smooth structure ¶
depends_on definition 13.48 Differentiable manifold ¶
- 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 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 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 lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - 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 13.98 $k$-form ¶
      - depends_on definition 13.84 Tensor ¶
      - 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 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 theorem 13.59 Regular value theorem ¶
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
    - depends_on definition 13.99 $k$-vector ¶
      - depends_on definition 13.116 Hodge dual ¶
    - depends_on definition 13.74 Length of a curve ¶
      - depends_on definition 13.76 Arclength parametrisation ¶
      - 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 13.145 Affine connection ¶
      - depends_on definition 13.130 Distribution; involutive; integrable ¶
      - 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.534 Every orbit is an embedded copy of $G$ ¶
    - 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.535 Existence of a slice ¶ ↺
  - 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 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 23.15 Orthogonal Hamiltonian ¶
  - depends_on definition 13.119 Induced metric ¶
    - depends_on proposition 13.162 Curvature induced on the quadric ¶
      - depends_on theorem 48.3 The Robertson–Walker metric ¶
  - depends_on definition 13.120 Isometry ¶
    - depends_on corollary 13.31 Bending invariance ¶
    - depends_on definition 13.139 Killing vector ¶
      - depends_on definition 13.142 Killing tensor ¶
      - depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
      - depends_on proposition 13.163 Integration of the conformal Killing equation ¶
      - depends_on proposition 13.140 Killing's equation ¶
    - depends_on proposition 13.121 The isometry group ¶
  - depends_on definition 13.74 Length of a curve ¶ ↺
  - depends_on definition 13.118 Line element ¶
    - depends_on definition 13.74 Length of a curve ¶ ↺
    - depends_on proposition 30.12 The Green strain measures length exactly ¶
      - depends_on example 30.13 A rigid rotation forges a strain of $-\theta^{2}/2$ ¶
  - depends_on definition 13.149 Metric compatibility ¶
    - depends_on definition 13.142 Killing tensor ¶ ↺
    - depends_on example 13.144 The two Killing tensors every metric carries ¶
    - depends_on lemma 44.8 Palatini identity ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - depends_on proposition 44.30 The cosmological term ¶
      - depends_on proposition 44.31 de Sitter geometry ¶
    - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
      - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶
    - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
      - depends_on phenomenon 45.52 The shadow ¶
      - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
      - depends_on proposition 45.6 Radial equation and effective potential ¶
      - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
    - depends_on proposition 13.140 Killing's equation ¶ ↺
    - depends_on proposition 13.154 Symmetries of the curvature ¶
      - depends_on definition 13.111 Linearised curvature of a symmetric field ¶
      - depends_on theorem 44.23 Contracted Bianchi identity ¶
      - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
    - depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
    - depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶ ↺
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on definition 13.77 Curvature vector ¶
      - depends_on lemma 44.8 Palatini identity ¶ ↺
      - depends_on proposition 44.44 Harmonic-gauge reduction ¶
      - depends_on proposition 21.73 Free motion in flat spacetime, any coordinates ¶
      - depends_on proposition 13.140 Killing's equation ¶ ↺
      - depends_on proposition 13.158 The connection determined by vielbein and torsion ¶
      - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
      - depends_on theorem 45.1 Schwarzschild solution ¶
  - depends_on definition 13.153 Contractions ¶
    - depends_on definition 44.2 Einstein tensor ¶
      - depends_on postulate 44.3 Einstein field equations ¶
      - depends_on theorem 44.23 Contracted Bianchi identity ¶ ↺
      - depends_on theorem 44.33 Lovelock's uniqueness theorem; imported ¶
    - depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
      - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
    - depends_on lemma A.631 The normal–normal Ricci contraction ¶
      - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶ ↺
    - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on theorem A.641 The normal–normal bracket ¶
    - depends_on proposition 15.31 Radius and cosmological constant ¶
      - depends_on corollary 15.32 The expansion parameter is the cosmological constant ¶
      - depends_on example 15.64 The whole chapter at $D=4$ ¶
      - depends_on remark 15.33 Which branch nature is on ¶
    - depends_on theorem A.49 thm:app-eh-equivalence ¶
    - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶ ↺
  - depends_on definition 13.122 Vielbein ¶
    - depends_on proposition 13.123 Local Lorentz freedom ¶
      - depends_on theorem A.49 thm:app-eh-equivalence ¶ ↺
    - depends_on theorem 43.3 Equivalence of the two variable sets ¶
      - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶
    - depends_on theorem 13.156 Cartan structure equations ¶
      - depends_on proposition 44.13 First-order variation in vielbein-form variables ¶ ↺
      - depends_on proposition 13.157 Bianchi identities ¶
      - depends_on proposition 13.158 The connection determined by vielbein and torsion ¶ ↺
  - depends_on definition 13.114 Volume form ¶
    - depends_on definition 13.116 Hodge dual ¶ ↺
  - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
    - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
      - depends_on lemma A.630 Codazzi equation ¶
      - depends_on lemma A.631 The normal–normal Ricci contraction ¶ ↺
  - depends_on proposition 23.44 Relativistic Hamilton–Jacobi equation ¶
    - depends_on remark 23.46 Carter's constant ¶
    - depends_on remark 23.45 Why this is not an analogy ¶
  - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
- 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 proposition A.539 Universal property ¶ ↺
  - depends_on theorem 13.59 Regular value theorem ¶ ↺
- depends_on definition 13.54 Embedded submanifold ¶
  - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
  - depends_on definition 13.130 Distribution; involutive; integrable ¶ ↺
  - depends_on definition 13.58 Hypersurface ¶
    - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - depends_on example A.294 The sphere, made explicit ¶
      - depends_on example 13.87 The $2$-sphere ¶ ↺
    - depends_on theorem 13.59 Regular value theorem ¶ ↺
  - depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
  - depends_on theorem 13.67 Quotient manifold theorem ¶ ↺
  - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶ ↺
- 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.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Atlas	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2509
`depends_on`	←	Uniqueness of the smooth structure	declared	appendices/A-long-proofs.tex:26318
`depends_on`	←	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2523