Differential geometry Geodesics Concept
Geodesic
Summary
A geodesic on a Riemannian manifold is a curve whose velocity is parallel along itself for the Levi-Civita connection. So it has zero acceleration within the manifold. Geodesics generalise straight lines: short pieces of a geodesic are the shortest curves between their endpoints. They are solutions of a second-order ordinary differential equation, so a geodesic is determined by its starting point and starting velocity. Geodesics are the basic tool for studying distance, curvature and topology of Riemannian manifolds.
Where this leads
32 pages build on this one.
- Exponential map
- Hartman's uniqueness theorem for harmonic maps
- Cartan–Hadamard theorem
- Preissmann's theorem
- Projective connection
- Hyperbolic triangle
- Arthur–Selberg trace formula
- Selberg trace formula
- Path space of a manifold
- Quantum ergodicity
- Quantum unique ergodicity conjecture
- Bott's Morse-theoretic proof of periodicity
- Myers's theorem
- Rauch comparison theorem
- Ultraparallel theorem
- Toponogov's theorem
- Horocycle
- Hopf–Rinow theorem
- Cheeger–Gromoll splitting theorem
- Poincaré half-plane model
- Fermat's principle
- Closed geodesic
- Geodesic flow
- Conjugate points
- Penrose–Hawking singularity theorems
- Cut locus
- Hilbert's fourth problem
- Jacobi field
- Morse index theorem for geodesics
- Synge's theorem
- First and second variation of arc length
- Maupertuis's principle
Further reading
About this page
| Tag | NSYMQMHPDN |
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| URI | m2://NSYMQMHPDN |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53C22 |
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Used by
- Exponential map
- Hartman's uniqueness theorem for harmonic maps
- Preissmann's theorem
- Projective connection
- Hyperbolic triangle
- Selberg trace formula
- Path space of a manifold
- Bott's Morse-theoretic proof of periodicity
- Rauch comparison theorem
- Ultraparallel theorem
- Toponogov's theorem
- Horocycle
- Hopf–Rinow theorem
- Cheeger–Gromoll splitting theorem
- Poincaré half-plane model
- Fermat's principle
- Closed geodesic
- Geodesic flow
- Conjugate points
- Penrose–Hawking singularity theorems
- Cut locus
- Hilbert's fourth problem
- Jacobi field
- Morse index theorem for geodesics
- Synge's theorem
- First and second variation of arc length
- Maupertuis's principle