definition 13.45 Differentiable map on a topological space

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2404 · p. 481

ground object -- no derivation owed

Rests on

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

Supports

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 13.53 Immersion, submersion, embedding ¶
    - 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 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 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 proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
      - depends_on proposition A.535 Existence of a slice ¶
    - depends_on lemma A.538 Submersions have smooth local sections ¶
      - depends_on proposition A.539 Universal property ¶
    - 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 remark 24.35 What non-squeezing does and does not say about nature ¶
      - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
      - depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
  - depends_on theorem 13.59 Regular value theorem ¶
    - 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 lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on proposition A.565 Freeness makes every value regular ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on proposition A.547 The components of the level set are the leaves of an integrable distribution ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on lemma A.555 The actions are well defined ¶
    - 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 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 proposition A.532 Separation and countability of the quotient ¶ ↺
  - depends_on theorem 13.67 Quotient manifold theorem ¶ ↺
- depends_on lemma A.538 Submersions have smooth local sections ¶ ↺
- depends_on proposition A.539 Universal property ¶ ↺
- depends_on theorem 13.59 Regular value theorem ¶ ↺
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 definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.43 Momentum map ¶
      - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
    - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶
    - 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 theorem A.68 Darboux ¶
    - depends_on definition 13.105 Closed form ¶
      - depends_on definition A.67 Symplectic manifold ¶ ↺
      - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on lemma 13.107 Poincaré lemma ¶
      - depends_on proposition 13.113 Product of a closed and an exact form ¶
      - depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
    - depends_on definition 13.106 Exact form ¶
      - depends_on example 13.110 Closed but not exact: the angle form ¶
      - depends_on lemma 13.107 Poincaré lemma ¶ ↺
      - depends_on proposition 13.113 Product of a closed and an exact form ¶ ↺
      - depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶ ↺
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on corollary 13.134 Frobenius for a Pfaffian system ¶
      - depends_on definition 13.105 Closed form ¶ ↺
      - depends_on definition 13.106 Exact form ¶ ↺
      - depends_on definition 25.21 First-order action in general coordinates ¶
      - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      - depends_on lemma A.312 The computation in one chart ¶
      - depends_on proposition 13.128 Cartan's magic formula ¶
      - depends_on theorem A.311 General Stokes theorem ¶
      - depends_on theorem 13.156 Cartan structure equations ¶
    - depends_on definition 13.116 Hodge dual ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.103 Exterior derivative ¶ ↺
      - depends_on definition 13.114 Volume form ¶
      - depends_on definition 24.20 Liouville volume ¶
      - depends_on lemma A.301 Transformation of the top form ¶
      - depends_on theorem 13.156 Cartan structure equations ¶ ↺
    - 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 A.304 Integral over a chart ¶
      - depends_on lemma A.301 Transformation of the top form ¶ ↺
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 13.99 $k$-vector ¶
      - depends_on definition 13.116 Hodge dual ¶ ↺
    - 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 definition 13.120 Isometry ¶
      - depends_on definition 13.74 Length of a curve ¶
      - depends_on definition 13.118 Line element ¶
      - depends_on definition 13.149 Metric compatibility ¶
      - depends_on definition 13.153 Contractions ¶
      - depends_on definition 13.122 Vielbein ¶
      - depends_on definition 13.114 Volume form ¶ ↺
      - depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      - depends_on proposition 23.44 Relativistic Hamilton–Jacobi equation ¶
      - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
    - 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 13.99 $k$-vector ¶ ↺
- depends_on definition 13.74 Length of a curve ¶ ↺
- 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 32.29 Limit cycle ¶
      - depends_on definition 32.67 Strange attractor ¶
      - depends_on theorem 32.74 Oseledets, quoted ¶
    - depends_on definition 32.22 Bifurcation ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on definition 32.10 Fixed point ¶
      - depends_on definition 32.12 Linearization ¶
      - depends_on definition 32.19 Lyapunov function ¶
      - depends_on definition 32.11 Lyapunov stability ¶
    - depends_on definition 32.41 The horseshoe map ¶
      - depends_on theorem 32.42 The horseshoe is a full shift, quoted ¶
    - depends_on definition 32.83 The logistic map ¶
    - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on definition 32.91 The circle map ¶
      - depends_on example 32.70 The Hénon map ¶
      - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
      - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - depends_on proposition 32.4 The flow is a one-parameter group ¶
      - depends_on corollary 32.5 Trajectories do not cross ¶
    - depends_on theorem 32.6 Evolution of phase volume ¶
      - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
      - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition A.625 Four-dimensional extrinsic curvature ¶
      - depends_on lemma A.630 Codazzi equation ¶
      - depends_on lemma A.628 Gauss equation, arbitrary signature ¶
      - depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    - depends_on definition A.624 Projector and induced metric ¶
    - depends_on definition 13.77 Curvature vector ¶
      - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
    - depends_on definition 13.149 Metric compatibility ¶ ↺
    - depends_on definition 13.146 Parallel transport and autoparallels ¶
      - depends_on definition 13.149 Metric compatibility ¶ ↺
      - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
      - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
      - depends_on remark 29.62 The same rate read as a holonomy ¶
      - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
    - depends_on definition 13.147 Torsion ¶
      - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
      - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
      - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - depends_on theorem 13.156 Cartan structure equations ¶ ↺
    - 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 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 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 ¶ ↺

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:2410
`depends_on`	→	Coordinate system	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2410
`depends_on`	←	Smooth map between manifolds	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2557
`depends_on`	←	Vector on a manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3916