definition 13.120 Isometry

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5538 · p. 520

ground object -- no derivation owed

Rests on

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 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.84 Tensor ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 13.81 Vector on a manifold ¶ ↺

Supports

depends_on corollary 13.31 Bending invariance ¶
depends_on definition 13.139 Killing vector ¶
- depends_on definition 13.142 Killing tensor ¶
  - depends_on example 13.144 The two Killing tensors every metric carries ¶
  - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
- depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶
  - 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 ¶
- depends_on proposition 13.163 Integration of the conformal Killing equation ¶
- depends_on proposition 13.140 Killing's equation ¶
  - depends_on example 13.144 The two Killing tensors every metric carries ¶ ↺
  - depends_on proposition 14.69 The de~Sitter algebras are isometry algebras ¶ ↺
  - depends_on proposition 14.59 The Killing fields of a flat pseudo-Euclidean space ¶ ↺
  - depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
    - depends_on phenomenon 45.52 The shadow ¶
      - depends_on phenomenon 50.4 A ring of light at event-horizon scale ¶
    - depends_on proposition 48.4 Peculiar momentum decays with the expansion ¶
    - depends_on proposition 45.6 Radial equation and effective potential ¶
      - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on proposition 45.8 Orbit equation ¶
      - depends_on proposition 45.21 Radial plunge: the cycloid ¶
    - depends_on theorem 13.143 Killing tensors and geodesic invariants ¶ ↺
  - depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
  - depends_on theorem 13.160 Maximal symmetry forces constant curvature ¶
    - depends_on proposition 44.31 de Sitter geometry ¶
    - 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 48.3 The Robertson–Walker metric ¶
      - depends_on phenomenon 48.12 The relic microwave background ¶
      - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
      - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
      - depends_on proposition 48.7 Luminosity distance ¶
depends_on proposition 13.121 The isometry group ¶

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`	→	Metric tensor of signature $(p,q)$	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5545
`depends_on`	←	Bending invariance	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1584
`depends_on`	←	Killing vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:6293
`depends_on`	←	The isometry group	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5551