## Definition

Let be a Riemannian manifold.

A vector field along a curve is termed a **Jacobi field** if it satisfies the following equation:

where is the tangent vector field along the curve.

The above is a second-order differential equations called the Jacobi equation.

## Facts

Jacobi fields are precisely the null space of the positive semidefinite quadratic form which is defined as:

where are variations with variation vector field .