theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2020 · p. 476

Rests on

depends_on equation 13.111 eq:mfd-egregium-liouville ¶
depends_on proposition 13.40 Parallel transport is an isometry of the tangent plane ¶
- 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 13.8 Curve ¶ ↺
  - depends_on definition 13.18 Surface ¶
- depends_on equation 13.126 eq:mfd-campoparalelo ¶
- depends_on equation 13.129 eq:mfd-geodesic-parallelism ¶
- proves proof ch:11-manifolds-tensors-curvature@proof-6 ¶
depends_on theorem 7.131 Green ¶
- depends_on definition 7.127 Simple regions ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof ch:05-real-analysis@proof-77 ¶
proves proof ch:11-manifolds-tensors-curvature@proof-7 ¶

Supports

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`	→	eq:mfd-egregium-liouville	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2040
`depends_on`	→	Parallel transport is an isometry of the tangent plane	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2040
`depends_on`	→	Green	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2040
`depends_on`	←	The sphere, the solid angle, and the pole	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2180
`proves`	←	ch:11-manifolds-tensors-curvature@proof-7	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2044