Let $f$ be a smooth, real valued function on a complete Riemannian manifold $(M,g)$ and $x_0\in M$. Define the function $\tilde f=f\circ Exp_{x_0}:T_{x_0}M\to \mathbb R$.
What is the Hessian of $\tilde f$, in terms of that of $f$ and characteristics of $(M,g)$?
Reason for this question: I am looking for a simple sufficient condition (ideally depending on bounds on the curvature of $(M,g)$) on $f$ so $\tilde f$ is convex.