# Geodesic for a linear connection

## 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:

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

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