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}.

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