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)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)