# Is the inverse of the exponential map smooth everywhere on a Complete Riemannian manifold

Let $(M, g)$ be a complete, smooth Riemannian manifold. $\forall p \in M, \ \exists U \subset M, \ p \in U ,$ s.t $exp^{-1}_{p}|_{U}$ is a diffeomorphism, and so $exp^{-1}_{p}$ is smooth on $U$.

My question is whether this property holds globally on a complete riemannian manifold. Is the inverse exponential map $exp^{-1}_{p} : M \rightarrow T_pM$ a smooth map for all $p \in M$ ?

If not, under what additional assumptions would this be true ?

• Iff there are not conjugate points and the manifold is simply connected. Exercise: Work out the case $M=S^1$. – Moishe Kohan Sep 21 '17 at 15:49
• $\exp _p ^{-1}$ is not defined unless $\ exp _p$ is a diffeomorphism, in particular $M$ must be diffeomorphic to $\bf R^n$ – Thomas Sep 22 '17 at 5:09