in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.
They are named after Carl Jacobi.
Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics
, 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
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
These correspond respectively to the following families of reparametrizations:
can be represented in a unique way as a sum
is a linear combination of trivial Jacobi fields and
then corresponds to the same variation of geodesics as
On a unit sphere, the geodesics through the North pole are great circles.
The most interesting information is just that Instead, we can consider the derivative with respect to
: 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.
and complete this to get an orthonormal basis
Parallel transport it to get a basis
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
with parallel orthonormal frame