= Radial distance in a normal neighbourhood
{title2=$r(q)=|\exp_p^{-1}(q)|$}
In a <normal neighbourhood> $U=\exp_p(V)$ with $V$ star-shaped, define $r(q)=|\exp_p^{-1}(q)|$. Away from $p$, the <Gauss lemma> gives $\nabla r=(d\exp_p)_v(v/|v|)$ for $v=\exp_p^{-1}(q)$. Thus $|\nabla r|=1$ and this unit <vector field> is tangent to unit-speed radial <geodesics>. The function $r$ is not smooth at $p$, but $r^2/2$ is smooth and $\nabla(r^2/2)=(d\exp_p)_v v$ extends by zero. On a sufficiently small normal ball, $r$ equals the <Riemannian distance> from $p$.
// Target: riemannian-geometry.bigb
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