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

From Diffgeom

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Equivalently, we say that <math>\gamma</math> is a geodesic if: | Equivalently, we say that <math>\gamma</math> is a geodesic if: | ||

− | <math>\frac{D}{dt}(\gamma'(t) | + | <math>\frac{D}{dt}(\gamma'(t)) = 0 \forall t</math> |

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

## Latest 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:

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