proposition 13.78 The curvature vector is orthogonal to the tangent

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3716 · p. 498

Rests on

depends_on definition 13.77 Curvature vector ¶
- depends_on definition 13.76 Arclength parametrisation ¶
  - depends_on definition 13.73 Admissible change of parameter ¶
  - depends_on definition 13.74 Length of a curve ¶
    - depends_on definition 13.118 Line element ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- depends_on definition 13.145 Affine connection ¶
  - depends_on definition 13.90 Mixed tensor ¶
    - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.149 Metric compatibility ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
    - depends_on definition 13.146 Parallel transport and autoparallels ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.147 Torsion ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
depends_on definition 13.149 Metric compatibility ¶ ↺
depends_on equation 13.170 eq:mfd-unit-tangent-intrinsic ¶
proves proof ch:11-manifolds-tensors-curvature@proof-15 ¶

Supports

depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Curvature vector	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3719
`depends_on`	→	Metric compatibility	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3719
`depends_on`	→	eq:mfd-unit-tangent-intrinsic	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3719
`depends_on`	←	Frenet–Serret equations, covariant form	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3768
`proves`	←	ch:11-manifolds-tensors-curvature@proof-15	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3723