Let $(M,g)$ be an $m$-dimensional complete Riemannian manifold with bounded sectional curvatures and $d(\cdot,\cdot)$ be the geodesic distance function. Suppose $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.  Define
$$
g(t)=\int_M f(t,x)e^{-f(t,x)}dx,
$$
where
$$
f(t,x)=d^2(Exp_\alpha(tv),x).
$$
Question: is it possible to show that **$g$ is second-order differentiable at $t=0$?**