definition 13.19 Parameter curves

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:926 · p. 461

ground object -- no derivation owed

Rests on

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 definition 13.20 Unit normal vector of a surface ¶
- depends_on definition 13.28 Normal curvature vector of a surface ¶
  - depends_on definition 13.29 Gaussian curvature ¶ ↺
  - depends_on definition 13.36 Geodesic curvature vector ¶
    - depends_on definition 13.37 Geodesic lines ¶ ↺
- depends_on definition 13.25 Second fundamental coefficients ¶
  - depends_on definition 13.29 Gaussian curvature ¶ ↺
  - depends_on definition 13.27 Riemann symbols ¶ ↺
  - depends_on proposition A.493 A harmonic conformal map is a minimal surface ¶
- depends_on definition 13.24 Second fundamental form ¶
  - depends_on definition 13.25 Second fundamental coefficients ¶ ↺

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`	→	Curve	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:933
`depends_on`	→	Surface	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:933
`depends_on`	←	First fundamental coefficients	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:996
`depends_on`	←	Unit normal vector of a surface	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:952