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:

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

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 $W_{i}$.