Differential geometry Riemannian and pseudo-Riemannian manifolds Concept
Levi-Civita connection
Summary
The Levi-Civita connection of a pseudo-Riemannian manifold is the affine connection that has no torsion and preserves the metric. By the fundamental theorem of Riemannian geometry there is exactly one such connection. It is the connection used to define geodesics, parallel transport and curvature of a Riemannian manifold.
Where this leads
17 pages build on this one.
- Berger's classification of holonomy groups
- Geodesic
- Hyperkähler manifold
- Gauss–Codazzi equations
- Fundamental theorem of Riemannian geometry
- Riemann curvature tensor
- Cotton tensor
- Spin(7)-manifold
- Weitzenböck identity
- Bochner's formula
- Information geometry
- Chern–Gauss–Bonnet theorem
- Laplace–Beltrami operator
- Killing vector field
- G2 manifold
- Quaternion-Kähler manifold
- Theorema Egregium
Further reading
About this page
| Tag | CAZYGE1T4E |
|---|---|
| URI | m2://CAZYGE1T4E |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53B05, 53B20 |
| Specializes | Affine connection |
| Same as | |
| Changed | 2 November 2026 |
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What links here
Used by
- Berger's classification of holonomy groups
- Geodesic
- Hyperkähler manifold
- Gauss–Codazzi equations
- Fundamental theorem of Riemannian geometry
- Riemann curvature tensor
- Cotton tensor
- Spin(7)-manifold
- Weitzenböck identity
- Bochner's formula
- Information geometry
- Chern–Gauss–Bonnet theorem
- Laplace–Beltrami operator
- Killing vector field
- G2 manifold
- Quaternion-Kähler manifold