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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^2J}{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.