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Geodesic for a linear connection

817 bytes added, 21:13, 6 January 2012
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 ..."

===Given data===

* A connected [[differential manifold]] <math>M</math> with [[tangent bundle]] denoted by <math>TM</math>.
* A [[defining ingredient::linear connection]] <math>\nabla</math> for <math>M</math>.

===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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