Differential geometry Riemannian and pseudo-Riemannian manifolds Concept
Riemannian metric
Summary
A Riemannian metric on a smooth manifold is a smoothly varying inner product on each tangent space. It lets one measure lengths of tangent vectors and angles between them, and from these the lengths of curves, distances, areas and volumes. It is a symmetric positive definite tensor field of type . Every smooth manifold admits Riemannian metrics, and the choice of one is the starting point of Riemannian geometry.
Where this leads
32 pages build on this one.
- Loewner's torus inequality
- Seiberg–Witten invariants
- Riemannian manifold
- Spectral decomposition theorem
- Morse homology
- Nash embedding theorems
- Theorem of the three geodesics
- Ebin's slice theorem
- Quantum ergodicity
- Donaldson theory
- Quantum unique ergodicity conjecture
- Information geometry
- Yamabe flow
- Hermitian manifold
- Yang–Mills equations
- Gromov's systolic inequality
- Ultraparallel theorem
- Hyperbolic set
- Horocycle
- Riemannian distance
- Birkhoff min-max method
- Poincaré half-plane model
- Axiom A
- Newlander–Nirenberg theorem
- Geodesic flow
- Anosov diffeomorphism
- Bangert–Franks theorem on closed geodesics on the sphere
- Stability conjecture
- Conformal geometry
- Christoffel symbols
- Manifold of smooth mappings
- Ricci flow
Further reading
About this page
| Tag | Q61VFHCH2J |
|---|---|
| URI | m2://Q61VFHCH2J |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53B20, 53C20 |
| Specializes | Tensor field |
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Definitions
Used by
- Loewner's torus inequality
- Seiberg–Witten invariants
- Riemannian manifold
- Morse homology
- Nash embedding theorems
- Theorem of the three geodesics
- Ebin's slice theorem
- Donaldson theory
- Information geometry
- Yamabe flow
- Hermitian manifold
- Yang–Mills equations
- Gromov's systolic inequality
- Riemannian distance
- Birkhoff min-max method
- Poincaré half-plane model
- Newlander–Nirenberg theorem
- Bangert–Franks theorem on closed geodesics on the sphere
- Conformal geometry
- Christoffel symbols
- Manifold of smooth mappings
- Ricci flow