definition 13.81 Vector on a manifold

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3910 · p. 500

ground object -- no derivation owed

Rests on

depends_on definition 13.71 Differentiable curve ¶
- depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
depends_on definition 13.45 Differentiable map on a topological space ¶
- depends_on definition 13.44 Atlas ¶ ↺
- depends_on definition 13.43 Coordinate system ¶ ↺
depends_on definition 13.48 Differentiable manifold ¶ ↺

Supports

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 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on example 24.29 A volume-preserving squeeze ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
  - 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 corollary A.69 No local invariants ¶
      - depends_on example A.80 The phase space of one particle in space ¶
      - depends_on theorem 24.12 Darboux ¶
  - depends_on definition 13.105 Closed form ¶
    - depends_on definition A.67 Symplectic manifold ¶ ↺
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      - depends_on definition 24.14 Hamiltonian vector field ¶
      - depends_on definition 24.20 Liouville volume ¶
      - depends_on definition 24.53 Real polarization and polarized sections ¶
      - depends_on definition 24.49 Prequantum datum ¶
      - depends_on definition 24.17 Symplectomorphism ¶ ↺
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
      - depends_on theorem 24.12 Darboux ¶ ↺
      - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      - depends_on theorem 24.38 Symplectic foliation, quoted ¶
      - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on lemma 13.107 Poincaré lemma ¶
      - depends_on proposition 13.113 Product of a closed and an exact form ¶
      - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
    - 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 example 13.110 Closed but not exact: the angle form ¶
      - depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
  - 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 theorem 25.22 Faddeev–Jackiw equations and brackets ¶
    - 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 lemma A.77 Poincaré lemma, converse form ¶ ↺
      - depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
      - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
    - 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 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 13.116 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 13.116 Hodge dual ¶ ↺
    - depends_on definition 24.20 Liouville volume ¶ ↺
    - depends_on lemma A.301 Transformation of the top form ¶
      - depends_on definition A.302 Orientation ¶
      - depends_on lemma A.305 The chart integral is well defined ¶
    - 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.305 The chart integral is well defined ¶ ↺
    - 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 proposition 13.162 Curvature induced on the quadric ¶
    - depends_on definition 13.120 Isometry ¶
      - depends_on corollary 13.31 Bending invariance ¶
      - depends_on definition 13.139 Killing vector ¶
      - depends_on proposition 13.121 The isometry group ¶
    - 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 13.118 Line element ¶
      - depends_on definition 13.74 Length of a curve ¶ ↺
      - depends_on proposition 30.12 The Green strain measures length exactly ¶
    - depends_on 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 proposition 44.30 The cosmological term ¶
      - depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
      - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
      - depends_on proposition 13.140 Killing's equation ¶
      - depends_on proposition 13.154 Symmetries of the 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 13.153 Contractions ¶
      - depends_on definition 44.2 Einstein tensor ¶
      - depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
      - depends_on lemma A.631 The normal–normal Ricci contraction ¶
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on proposition 15.31 Radius and cosmological constant ¶
      - 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 43.3 Equivalence of the two variable sets ¶
      - depends_on theorem 13.156 Cartan structure equations ¶ ↺
    - depends_on definition 13.114 Volume form ¶ ↺
    - 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 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 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 corollary 13.64 The image of a constant-rank map, locally ¶
    - 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 13.58 Hypersurface ¶
      - depends_on example 13.61 The $2$-sphere in $\R^{3}$ ¶
      - 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 example 13.68 Why each hypothesis is there ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 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 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.536 A slice is a chart domain downstairs ¶
      - depends_on remark A.542 Where each hypothesis is spent ¶
  - 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.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 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 13.74 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 phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on definition 32.67 Strange attractor ¶
      - depends_on example 32.70 The Hénon map ¶
      - depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
      - depends_on phenomenon 32.68 Strange attractors ¶
    - depends_on theorem 32.74 Oseledets, quoted ¶
      - depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
  - depends_on definition 32.22 Bifurcation ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - depends_on definition 32.10 Fixed point ¶
    - depends_on definition 32.12 Linearization ¶
      - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on theorem 32.13 Linear stability ¶
    - depends_on definition 32.19 Lyapunov function ¶
      - depends_on theorem 32.20 Lyapunov's direct method ¶
    - depends_on definition 32.11 Lyapunov stability ¶
      - depends_on theorem 32.20 Lyapunov's direct method ¶ ↺
  - depends_on definition 32.41 The horseshoe map ¶
    - depends_on theorem 32.42 The horseshoe is a full shift, quoted ¶
      - depends_on example 32.80 The middle-thirds Cantor set ¶
      - depends_on proposition 32.43 What the shift does ¶
  - 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 proposition 32.93 The circle map loses invertibility at $K=1$ ¶
    - 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 phenomenon 32.87 Universal period doubling ¶
      - depends_on proposition 32.84 Fixed points and the first period doubling ¶
  - depends_on proposition 32.4 The flow is a one-parameter group ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶
      - depends_on definition 32.29 Limit cycle ¶ ↺
      - depends_on definition 32.38 Poincaré section and return map ¶ ↺
      - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
  - 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 definition 32.62 The Lorenz system ¶ ↺
    - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on phenomenon 32.68 Strange attractors ¶ ↺
    - 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 theorem 44.42 The constraint equations ¶
    - depends_on lemma A.628 Gauss equation, arbitrary signature ¶
      - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
      - depends_on theorem 44.42 The constraint equations ¶ ↺
    - 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 definition 13.153 Contractions ¶ ↺
      - depends_on example 21.74 Rindler coordinates ¶
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶ ↺
      - depends_on lemma 44.8 Palatini identity ¶ ↺
      - depends_on proposition 44.44 Harmonic-gauge reduction ¶
      - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶ ↺
      - depends_on proposition 13.140 Killing's equation ¶ ↺
      - depends_on proposition 13.162 Curvature induced on the quadric ¶ ↺
      - depends_on proposition 13.154 Symmetries of the curvature ¶ ↺
      - depends_on remark 30.16 Incompatibility is curvature ¶
      - depends_on theorem 13.156 Cartan structure equations ¶ ↺
    - 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 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 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Differentiable curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3916
`depends_on`	→	Differentiable map on a topological space	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3916
`depends_on`	→	Differentiable manifold	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3916
`depends_on`	←	Covector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3993
`depends_on`	←	Differential; pushforward	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2617
`depends_on`	←	$k$-vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4798
`depends_on`	←	Length of a curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3644
`depends_on`	←	Tensor	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4030
`depends_on`	←	Vector field	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3922
`depends_on`	←	The $2$-sphere	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:4203