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

• A connected differential manifold $M$ with tangent bundle denoted by $TM$.
• A linear connection $\nabla$ for $M$.
Consider a smooth curve $\gamma:[0,1] \to M$. Consider the connection along $\gamma$ induced by $\nabla$, and consider the transport along $\gamma$ for that connection. 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 $\gamma(t)$.