Examples
Consider a geodesic with parallel orthonormal frame, constructed as above.
- The vector fields along given by and are Jacobi fields.
- In Euclidean space (as well as for spaces of constant zero sectional curvature) Jacobi fields are simply those fields linear in .
- For Riemannian manifolds of constant negative sectional curvature, any Jacobi field is a linear combination of, and, where .
- For Riemannian manifolds of constant positive sectional curvature, any Jacobi field is a linear combination of, and, where .
- The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold.
- The Jacobi fields correspond to the geodesics on the tangent bundle (with respect to the metric on induced by the metric on ).
Read more about this topic: Jacobi Field
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)