Geodesic Hessian formula
= Geodesic Hessian formula
{title2=$(f\circ\gamma)^{\prime\prime}(0)=\operatorname{Hess}f(v,v)$}
For a <geodesic> $\gamma$ with initial velocity $v$, $(f\circ\gamma)^{\prime\prime}(0)=\operatorname{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an <orthonormal basis> of the tangent space gives $\Delta f(p)=-\sum_i(f\circ\gamma_i)^{\prime\prime}(0)$ for the nonnegative <Laplace-Beltrami operator>.