# Difference between revisions of "Jacobi field"

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==Definition== | ==Definition== | ||

− | + | Let <math>M</math> be a [[Riemannian manifold]]. | |

− | <math> | + | A vector field <math>J</math> along a curve <math>\omega:[0,1] \to M</math> is termed a '''Jacobi field''' if it satisfies the following equation: |

+ | |||

+ | <math>\frac{D^2J}{dt^2} + R(J,V)V = 0</math> | ||

+ | |||

+ | where <math>V</math> is the tangent vector field along the curve. | ||

The above is a second-order differential equations called the Jacobi equation. | 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 <math>E_{**}</math> which is defined as: | ||

+ | |||

+ | <math>E_{**} (W_1,W_2) = \frac{\partial^2 E (\overline{\alpha}(u_1,u_2)}{\partial u_1\partial u_2}</math> | ||

+ | |||

+ | where <math>u_i</math> are [[variation]]s with [[variation vector field]] <math>W_i</math>. |

## Revision as of 11:49, 5 August 2007

## 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 .