Jacobi field

Definition

Let $M$ be a Riemannian manifold.

A vector field $J$ along a curve $\omega :[0,1]\to M$ is termed a Jacobi field if it satisfies the following equation:

$\frac{D^{2}J}{d{t}^2}+R(J,V)V=0$

where $V$ 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 $E_{**}$ which is defined as:

$E_{**}(W_1,W_2)=\frac{{\partial }^{2}E(\overline{\alpha }(u_1,u_2)}{\partial u_1\partial u_2}$

where $u_i$ are variations with variation vector field $W_i$.