Differential geometry Curvature and topology Concept
Sectional curvature
Summary
The sectional curvature of a Riemannian manifold assigns a number to each -dimensional plane in each tangent space. It is the Gaussian curvature at of the surface swept out by the geodesics from tangent to that plane. The sectional curvatures determine the full Riemann curvature tensor, and bounds on them are the hypotheses of the main comparison theorems of global Riemannian geometry.
Where this leads
19 pages build on this one.
- Sacks–Uhlenbeck bubbling
- Hartman's uniqueness theorem for harmonic maps
- Cartan–Hadamard theorem
- Mostow rigidity theorem
- Preissmann's theorem
- Sphere theorem
- Eells–Sampson theorem
- Hyperbolic 3-manifold
- Soul theorem
- Hopf conjecture
- Quantum ergodicity
- Quantum unique ergodicity conjecture
- Rauch comparison theorem
- Toponogov's theorem
- Killing–Hopf theorem
- Margulis lemma
- Space form
- Geodesic flow
- Synge's theorem
Further reading
About this page
| Tag | 1X42JAA349 |
|---|---|
| URI | m2://1X42JAA349 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53B20, 53C20 |
| Same as |
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Used by
- Hartman's uniqueness theorem for harmonic maps
- Cartan–Hadamard theorem
- Mostow rigidity theorem
- Preissmann's theorem
- Sphere theorem
- Eells–Sampson theorem
- Hyperbolic 3-manifold
- Soul theorem
- Hopf conjecture
- Quantum unique ergodicity conjecture
- Rauch comparison theorem
- Toponogov's theorem
- Killing–Hopf theorem
- Margulis lemma
- Space form
- Geodesic flow
- Synge's theorem
Results about this
- Cartan–Hadamard theorem
- Preissmann's theorem
- Sphere theorem
- Soul theorem
- Hopf conjecture
- Rauch comparison theorem
- Toponogov's theorem
- Synge's theorem