proposition 13.162 Curvature induced on the quadric

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7152 · p. 537

Rests on

depends_on definition 13.119 Induced metric ¶
- 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 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
depends_on equation 13.83 eq:mfd-ecsgauss ¶
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
- 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 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on definition 13.147 Torsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
- depends_on proposition 7.105 Clairaut–Schwarz ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - 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-49 ¶

Supports

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 103.68 The sum of the neutrino masses is bounded ¶
  - 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
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Edges

type	direction	node	provenance	where
`depends_on`	→	Induced metric	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168
`depends_on`	→	eq:mfd-ecsgauss	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168
`depends_on`	←	The Robertson–Walker metric	declared	parts/05-general-relativity-cosmology/07-cosmology.tex:240
`proves`	←	ch:11-manifolds-tensors-curvature@proof-49	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7171