proposition 13.40 Parallel transport is an isometry of the tangent plane
open in the book ·
parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:1995
· p. 476
Rests on
-
depends_on
definition 13.21
First fundamental form
¶
- depends_on definition 13.11 Arc length ¶
- 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 ¶
Supports
-
depends_on
theorem 13.41
Holonomy equals the enclosed curvature; local
Gauss–Bonnet
¶
- 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 |
→ | First fundamental form | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2001 |
depends_on |
→ | eq:mfd-campoparalelo | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2001 |
depends_on |
→ | eq:mfd-geodesic-parallelism | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2001 |
depends_on |
← | Holonomy equals the enclosed curvature; local Gauss–Bonnet | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2040 |
proves |
← | ch:11-manifolds-tensors-curvature@proof-6 | declared | parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:2005 |