Definitions and Properties
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics with, then
is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic .
A vector field J along a geodesic is said to be a Jacobi field if it satisfies the Jacobi equation:
where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics describing the field (as in the preceding paragraph).
The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of and at one point of uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold.
As trivial examples of Jacobi fields one can consider and . These correspond respectively to the following families of reparametrisations: and .
Any Jacobi field can be represented in a unique way as a sum, where is a linear combination of trivial Jacobi fields and is orthogonal to, for all . The field then corresponds to the same variation of geodesics as, only with changed parameterizations.
Read more about this topic: Jacobi Field
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