Geodesic for a linear connection: Difference between revisions

Revision as of 21:17, 6 January 2012

Definition

Given data

A connected differential manifold $M$ with tangent bundle denoted by $TM$ .
A linear connection $\nabla$ for $M$ .

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): {\displaystyle \frac{D}{dt}(\gamma'(t)} = 0 \forall t}

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

@@ Line 8: / Line 8: @@
 ===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>.
+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>.
+Equivalently, we say that <math>\gamma</math> is a geodesic if:
+<math>\frac{D}{dt}(\gamma'(t)} = 0 \forall t</math>
+(with the derivative interpreted as a suitable one-sided derivative at the endpoints).