definition 13.77 Curvature vector

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3693 · p. 498

ground object -- no derivation owed

Rests on

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 13.48 Differentiable manifold ¶
      - depends_on definition 13.84 Tensor ¶
  - 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 13.48 Differentiable manifold ¶ ↺
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - 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 ¶

Supports

depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
- depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶

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`	→	Arclength parametrisation	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3705
`depends_on`	→	Affine connection	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3705
`depends_on`	→	Levi-Civita connection and contorsion	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3705
`depends_on`	←	The curvature vector is orthogonal to the tangent	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:3719