 Exponential map (Riemannian geometry)

For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_{p}M$, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map isEquivalently, ${\exp }_p(tv)={\gamma }_v(t)$ wherever both sides are defined.

In Riemannian geometry, the exponential map is a crucial concept that connects the local geometric properties of a Riemannian manifold to its global structure. Specifically, it describes how to move along geodesics (the generalization of straight lines to curved spaces) starting from a given point on the manifold.