Vipul: 2 revisions

2008-05-18T19:47:51Z

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Vipul at 11:49, 5 August 2007

2007-08-05T11:49:49Z

← Older revision		Revision as of 11:49, 5 August 2007
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	==Definition==		==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.		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>.

Vipul at 08:50, 22 June 2007

2007-06-22T08:50:22Z

New page

==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:

<math>D_TD_T(V) + R(T,V)T = 0</math>

The above is a second-order differential equations called the Jacobi equation.

Jacobi field - Revision history

Vipul: 2 revisions

Vipul at 11:49, 5 August 2007

Vipul at 08:50, 22 June 2007