equation 13.83 eq:mfd-ecsgauss

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1203

connected by a declared semantic edge

Rests on

No declared or derived dependency edges point away from this node yet.

Supports

depends_on definition 13.26 Christoffel symbols ¶
- depends_on definition 13.37 Geodesic lines ¶
- depends_on definition 13.27 Riemann symbols ¶
  - depends_on theorem 13.30 Theorema Egregium; Gauss, 1827 ¶
    - depends_on corollary 13.31 Bending invariance ¶
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`	←	Christoffel symbols	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1222
`depends_on`	←	Curvature induced on the quadric	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:7168