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 $V \in T_pM$, the exponential map at $p$, denoted $exp_p$, is defined as follows: for every $V\in T_pM$, there is a unique geodesic $\gamma_V: \mathbb{R} \to M$ with $\gamma_V(0) = p$ and $\gamma_V'(0) = V$. The exponential map $exp_p: T_pM \to M$ is then defined by $exp_p(V) = \gamma_V(1)$. To get $exp_p(tV)$, you follow the geodesic for a "time" $t$ instead of 1, so $exp_p(tV) = \gamma_{tV}(1)$.
In other words, the exponential map $exp_p(tV)$ 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.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: