Exponential map

Summary

Let $p$ be a point of a Riemannian manifold. The exponential map at $p$ sends $v \in T_pM$ to the point reached at time $1$ by the geodesic from $p$ with velocity $v$. It maps a small ball around $0$ in the tangent space diffeomorphically onto a neighbourhood of $p$. So it gives coordinates adapted to the geometry, the normal coordinates. It is central to the study of completeness, conjugate points and the cut locus. For a bi-invariant metric on a Lie group, it agrees at the identity with the exponential map of the Lie group.