Jacobi field Guide, Meaning , Facts, Information and Description
In Riemannian geometry, a Jacobi field is a certain type of vector field along a geodesic in a Riemannian manifold. Jacobi fields are one of the basic objects of study in Riemannian geometry; for the origin of the name, see Carl Jacobi.
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2 Motivating example 3 Solving the Jacobi Equation 4 Examples 5 References |
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics with , then
A field J is a Jacobi field if and only if it satisfies the Jacobi equation:
Definitions and properties
is a Jacobi field.
where D denotes the Levi-Civita connection, R the curvature tensor and .
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 define uniquely the Jacobi field. Further, the sum of Jacobi fields on a given geodesic is again a Jacobi field.
As trivial examples of Jacobi fields one can consider and . These correspond respectively to the following families of reparametrisations: and .
Any Jacobi field 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 parametrizations.
On a sphere, the geodesics through the North pole are great circles. Consider two such geodesics and with natural parameter, , separated by an angle . The geodesic distance is
Motivating example
Computing this requires knowing the geodesics. The most interesting information is just that
Instead, we can consider the derivative with respect to at :
Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.
Let and complete this to get an orthonormal basis at . Parallel transport it to get a basis all along .
This gives an orthonormal basis with . The Jacobi field is and thus
Solving the Jacobi Equation
and the Jacobi equation can be rewritten as a system
for each . This way we get a linear ordinary differential equation (ODE).
Since this ODE has smooth coefficients we have that solutions exist for all and are unique, given and , for all .
Consider a geodesic with parallel basis frame , , constructed as above.
In Euclidean space (as well as for spaces of constant zero curvature) Jacobi fields are simply those fields linear in .
For Riemannian manifolds of constant negative curvature , any Jacobi field is a linear combination of , and , where .
For Riemannian manifolds of constant positive curvature , any Jacobi field is a linear combination of , , and , where .Examples