proposition 14.59 The Killing fields of a flat pseudo-Euclidean space

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2390 · p. 568

Rests on

depends_on definition 13.139 Killing vector ¶
- depends_on definition 13.120 Isometry ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
- depends_on equation 13.267 eq:mfd-lie-def ¶
depends_on equation 14.86 eq:lie-sopq-gen ¶
depends_on proposition 13.140 Killing's equation ¶
- depends_on definition 13.139 Killing vector ¶ ↺
- depends_on definition 13.149 Metric compatibility ¶
  - depends_on definition 13.145 Affine connection ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.82 Vector field ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
- depends_on proposition 13.127 Component formulas ¶
  - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
  - depends_on definition 13.90 Mixed tensor ¶ ↺
  - depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.149 Metric compatibility ¶ ↺
  - depends_on definition 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶ ↺
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶
proves proof ch:12-lie-groups-fibre-bundles@proof-27 ¶

Supports

depends_on corollary A.344 The algebras $\mathfrak{so}(p,q)$ are semisimple ¶
- depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
- depends_on example A.359 The Lorentz algebra ¶
depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶

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`	→	Killing vector	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2411
`depends_on`	→	eq:lie-sopq-gen	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2411
`depends_on`	→	Killing's equation	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2411
`depends_on`	←	The algebras $\mathfrak{so}(p,q)$ are semisimple	declared	appendices/A-long-proofs.tex:16768
`depends_on`	←	The de~Sitter algebras are isometry algebras	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:3103
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-27	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:2414