# Difference between revisions of "Geodesic for a linear connection"

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− | Consider a [[smooth curve]] <math>\gamma:[0,1] \to M</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>. |

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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

- A connected differential manifold with tangent bundle denoted by .
- A linear connection for .

### Definition part

Consider a smooth curve . Let denote the connection along induced by , and consider the transport along for the connection . Then, we say that is a **geodesic** for if, under that transport, the tangent vector at gets transported, at time , to the tangent vector at .

Equivalently, we say that 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).