Difference between revisions of "Jacobi field"

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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:
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Let <math>M</math> be a [[Riemannian manifold]].
  
<math>D_TD_T(V) + R(T,V)T = 0</math>
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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:
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<math>\frac{D^2J}{dt^2} + R(J,V)V = 0</math>
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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.
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==Facts==
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Jacobi fields are precisely the null space of the positive semidefinite quadratic form <math>E_{**}</math> which is defined as:
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<math>E_{**} (W_1,W_2) = \frac{\partial^2 E (\overline{\alpha}(u_1,u_2)}{\partial u_1\partial u_2}</math>
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where <math>u_i</math> are [[variation]]s with [[variation vector field]] <math>W_i</math>.

Latest revision as of 19:47, 18 May 2008

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.