# Changes

## Geodesic for a linear connection

, 21:13, 6 January 2012
Created page with "==Definition== ===Given data=== * A connected differential manifold $M$ with tangent bundle denoted by $TM$. * A [[defining ingredient::linear ..."
==Definition==

===Given data===

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

===Definition part===

Consider a [[smooth curve]] $\gamma:[0,1] \to M$. Consider the [[connection along a curve|connection along]] $\gamma$ induced by $\nabla$, and consider the [[transport along a curve|transport along]] $\gamma$ for that connection. 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 $\gamma(t)$.