theorem 13.30 Theorema Egregium; Gauss, 1827

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1547 · p. 469

Rests on

depends_on definition 13.29 Gaussian curvature ¶
- depends_on definition 13.22 First fundamental coefficients ¶
  - depends_on definition 13.21 First fundamental form ¶
    - 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 13.18 Surface ¶
  - depends_on definition 13.19 Parameter curves ¶
    - depends_on definition 13.8 Curve ¶ ↺
    - depends_on definition 13.18 Surface ¶ ↺
- depends_on definition 13.28 Normal curvature vector of a surface ¶
  - depends_on definition 13.13 Normal curvature vector of a curve ¶
    - depends_on definition 13.12 Unit tangent vector ¶
      - depends_on definition 13.11 Arc length ¶ ↺
      - depends_on definition 13.9 Tangent vector to a curve ¶ ↺
  - depends_on definition 13.20 Unit normal vector of a surface ¶
    - depends_on definition 13.19 Parameter curves ¶ ↺
- depends_on definition 13.25 Second fundamental coefficients ¶
  - depends_on definition 13.24 Second fundamental form ¶
    - depends_on definition 13.20 Unit normal vector of a surface ¶ ↺
  - depends_on definition 13.20 Unit normal vector of a surface ¶ ↺
depends_on definition 13.27 Riemann symbols ¶
- depends_on definition 13.26 Christoffel symbols ¶
  - depends_on definition 13.23 Inverse metric tensor ¶
    - depends_on definition 13.22 First fundamental coefficients ¶ ↺
  - depends_on equation 13.83 eq:mfd-ecsgauss ¶
- depends_on definition 13.25 Second fundamental coefficients ¶ ↺
- depends_on equation 13.92 eq:mfd-ecscoda1 ¶
depends_on equation 13.95 eq:mfd-riemann-surface ¶
depends_on equation 13.88 eq:mfd-sc22 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-3 ¶

Supports

depends_on corollary 13.31 Bending invariance ¶

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`	→	Gaussian curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1552
`depends_on`	→	Riemann symbols	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1552
`depends_on`	→	eq:mfd-riemann-surface	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1552
`depends_on`	→	eq:mfd-sc22	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1552
`depends_on`	←	Bending invariance	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1584
`proves`	←	ch:11-manifolds-tensors-curvature@proof-3	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1555