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
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