Difference between revisions of "Geodesic for a linear connection"

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(Created page with "==Definition== ===Given data=== * A connected differential manifold <math>M</math> with tangent bundle denoted by <math>TM</math>. * A [[defining ingredient::linear ...")
 
(Definition part)
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===Definition part===
 
===Definition part===
  
Consider a [[smooth curve]] <math>\gamma:[0,1] \to M</math>. Consider the [[connection along a curve|connection along]] <math>\gamma</math> induced by <math>\nabla</math>, and consider the [[transport along a curve|transport along]] <math>\gamma</math> for that connection. Then, we say that <math>\gamma</math> is a '''geodesic''' for <math>\nabla</math> if, under that transport, the tangent vector <math>\gamma'(0)</math> at <math>\gamma(0)</math> gets transported, at time <math>t</math>, to the tangent vector <math>\gamma'(t)</math> at <math>\gamma(t)</math>.
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Consider a [[smooth curve]] <math>\gamma:[0,1] \to M</math>. Let <math>D/dt</math> denote the [[connection along a curve|connection along]] <math>\gamma</math> induced by <math>\nabla</math>, and consider the [[transport along a curve|transport along]] <math>\gamma</math> for the connection <math>D/dt</math>. Then, we say that <math>\gamma</math> is a '''geodesic''' for <math>\nabla</math> if, under that transport, the tangent vector <math>\gamma'(0)</math> at <math>\gamma(0)</math> gets transported, at time <math>t</math>, to the tangent vector <math>\gamma'(t)</math> at <math>\gamma(t)</math>.
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Equivalently, we say that <math>\gamma</math> is a geodesic if:
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<math>\frac{D}{dt}(\gamma'(t)} = 0 \forall t</math>
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(with the derivative interpreted as a suitable one-sided derivative at the endpoints).

Revision as of 21:17, 6 January 2012

Definition

Given data

Definition part

Consider a smooth curve \gamma:[0,1] \to M. Let D/dt denote the connection along \gamma induced by \nabla, and consider the transport along \gamma for the connection D/dt. Then, we say that \gamma is a geodesic for \nabla if, under that transport, the tangent vector \gamma'(0) at \gamma(0) gets transported, at time t, to the tangent vector \gamma'(t) at \gamma(t).

Equivalently, we say that \gamma is a geodesic if:

Failed to parse (syntax error): \frac{D}{dt}(\gamma'(t)} = 0 \forall t

(with the derivative interpreted as a suitable one-sided derivative at the endpoints).