Geodesic for a linear connection

Definition

Given data

• A connected differential manifold $M$ with tangent bundle denoted by $TM$.
• A linear connection $\mathrm{\nabla }$ for $M$.

Definition part

Consider a smooth curve $\gamma :[0,1]\to M$. Let $D/dt$ denote the connection along $\gamma$ induced by $\mathrm{\nabla }$, and consider the transport along $\gamma$ for the connection $D/dt$. Then, we say that $\gamma$ is a geodesic for $\mathrm{\nabla }$ if, under that transport, the tangent vector ${\gamma }^{\prime }(0)$ at $\gamma (0)$ gets transported, at time $t$, to the tangent vector ${\gamma }^{\prime }(t)$ at $\gamma (t)$.

Equivalently, we say that $\gamma$ is a geodesic if:

$\frac{D}{dt}({\gamma }^{\prime }(t))=0\mathrm{\forall }t$

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