## Smoothness of distance function

Suppose ${M}$ is a complete Riemannian manifold. ${p\in M}$ define ${\rho(x)=d(x,p)}$. Obviously ${\rho(x)}$ is continuous, what can we say about the smoothness of ${\rho}$.

(1) ${\rho(x)}$ is not ${C^1}$ near ${p}$;

(2) If ${M}$ is compact, then ${\rho(x)}$ is not ${C^1}$ in ${M\backslash \{p\}}$

It seems that ${\rho(x)}$ is not so smooth, let us consider ${\rho^2(x)}$

(3) ${\rho^2(x)}$ is smooth at neighborhood ${U}$ of ${p}$ and ${D^2\rho^2}$ is positive definite in ${U}$.

(4) If ${M}$ is simply connected complete manifold with ${Sec_M\leq 0}$, then ${\rho^2}$ is ${C^\infty}$ on whole ${M}$ and ${D^2\rho^2}$ is positive definite on ${M}$.