definition 13.119 Induced metric

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5523 · 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 proposition 13.162 Curvature induced on the quadric ¶
- depends_on theorem 48.3 The Robertson–Walker metric ¶
  - depends_on phenomenon 48.12 The relic microwave background ¶
    - depends_on phenomenon 48.13 The anisotropy spectrum of the relic radiation ¶
      - depends_on phenomenon 48.18 The Hubble tension ¶
    - tests experiment The Cosmic Microwave Background ¶
      - cites (method) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶
      - cites (evidence) derived bib 24 Detection ofB-Mode Polarization at Degree Angular Scales by BICEP2 ¶ ↺
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶
      - cites (evidence) derived bib 10 Joint Analysis of BICEP2/Keck ArrayandPlanckData ¶ ↺
      - cites (method) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (context) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib 15 Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season ¶ ↺
      - cites (evidence) derived bib Planck 2018 results. VI. Cosmological parameters ¶
      - cites (method) derived bib Planck 2018 results. VI. Cosmological parameters ¶ ↺
      - … 109 more
  - depends_on phenomenon 48.1 The Hubble–Lemaître law ¶
  - depends_on phenomenon 51.6 Acoustic peaks, and a spatially flat universe ¶
    - depends_on phenomenon 51.7 E-mode polarization, and no primordial B mode ¶
  - depends_on proposition 48.7 Luminosity distance ¶
    - depends_on corollary 48.8 Angular-diameter distance and the duality relation ¶
      - depends_on phenomenon 48.15 A standard ruler in the galaxy distribution ¶
    - depends_on phenomenon 48.17 The expansion is accelerating ¶
    - depends_on phenomenon 48.18 The Hubble tension ¶ ↺

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:5533
`depends_on`	←	Curvature induced on the quadric	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168