definition 13.21 First fundamental form

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:979 · p. 462

ground object -- no derivation owed

Rests on

depends_on definition 13.11 Arc length ¶
- depends_on definition 13.8 Curve ¶
- depends_on definition 13.9 Tangent vector to a curve ¶
  - depends_on definition 13.8 Curve ¶ ↺
depends_on definition 13.18 Surface ¶

Supports

depends_on definition 13.22 First fundamental coefficients ¶
- depends_on definition A.476 Disc-type surfaces and the two functionals ¶
  - depends_on definition A.481 Douglas' boundary functional ¶
    - depends_on definition A.486 The normalised admissible class ¶
    - depends_on lemma A.484 Lower semicontinuity of the Douglas functional ¶
    - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶
    - depends_on theorem A.482 Douglas' identity ¶
  - depends_on lemma A.479 Conformal invariance of the Dirichlet integral ¶
    - depends_on lemma A.487 The normalisation costs nothing ¶
      - depends_on theorem A.492 Douglas' conformality theorem; quoted ¶ ↺
    - depends_on lemma A.485 Conformal automorphisms of the disc ¶
      - depends_on definition A.486 The normalised admissible class ¶ ↺
      - depends_on lemma A.487 The normalisation costs nothing ¶ ↺
  - depends_on lemma A.489 Courant–Lebesgue ¶
    - depends_on theorem A.490 Equicontinuity of the normalised class ¶
  - depends_on proposition A.478 $D\ge A$, with equality exactly for conformal maps ¶
- depends_on definition 13.29 Gaussian curvature ¶
  - depends_on theorem 13.30 Theorema Egregium; Gauss, 1827 ¶
    - depends_on corollary 13.31 Bending invariance ¶
- depends_on definition 13.23 Inverse metric tensor ¶
  - 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 proposition 13.40 Parallel transport is an isometry of the tangent plane ¶
- depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
  - depends_on example 13.42 The sphere, the solid angle, and the pole ¶

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`	→	Arc length	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:983
`depends_on`	→	Surface	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:983
`depends_on`	←	First fundamental coefficients	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:996
`depends_on`	←	Parallel transport is an isometry of the tangent plane	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2001