Jacobi field

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In Riemannian geometry, a Jacobi field is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.

Definitions and properties[edit]

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.

Motivating example[edit]

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

Computing this requires knowing the geodesics. The most interesting information is just that

, for any .

Instead, we can consider the derivative with respect to at :

Notice that we still detect the intersection of the geodesics at . Notice further that to calculate this derivative we do not actually need to know

,

rather, all we need do is solve the equation

,

for some given initial data.

Jacobi fields give a natural generalization of this phenomenon to arbitrary Riemannian manifolds.

Solving the Jacobi equation[edit]

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 can be written in co-ordinates in terms of this basis as and thus

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 .

Examples[edit]

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 ).

See also[edit]

References[edit]