Jacobi field: Difference between revisions

Latest revision as of 19:47, 18 May 2008

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.

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 $W_{i}$ .

@@ Line 1: / Line 1: @@
 ==Definition==
-A vector field <math>V</math> along a curve <math>\gamma:[0,1] \to M</math> is termed a '''Jacobi field''' if it satisfies the following equation, where <math>T</math> is the tangent field for the curve and <math>D</math> denotes covariant derivative:
+Let <math>M</math> be a [[Riemannian manifold]].
-<math>D_TD_T(V) + R(T,V)T = 0</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.
+==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>.