# Geodesic for a linear connection

From Diffgeom

## 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:

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