Exponential map (Riemannian manifolds)

The exponential map is a way of moving from the tangent space at a point on a manifold to the manifold itself. Given a Riemannian manifold $(M,g)$, a point $p\in M$, and a tangent vector $V\in T_{p}M$, the exponential map at $p$, denoted $ex{p}_p$, is defined as follows: for every $V\in T_{p}M$, there is a unique geodesic ${\gamma }_V:\mathbb{R}\to M$ with ${\gamma }_V(0)=p$ and ${\gamma }_V^{\prime }(0)=V$. The exponential map $ex{p}_p:T_{p}M\to M$ is then defined by $ex{p}_p(V)={\gamma }_V(1)$. To get $ex{p}_p(tV)$, you follow the geodesic for a "time" $t$ instead of 1, so $ex{p}_p(tV)={\gamma }_{tV}(1)$.

In other words, the exponential map $ex{p}_p(tV)$ for $t\in \mathbb{R}$ is the point reached by traveling along the geodesic in $M$ starting at $p$ in the direction of $V$ for "time" $t$.

It let us define Riemann normal coordinates.

The exponential map for every $p$ and for every $V$ is controlled by the geodesic flow.