# Geodesic for a linear connection

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## Definition

### Given data

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

### Definition part

Consider a smooth curve . Consider the connection along induced by , and consider the transport along for that 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 .