= Geodesic trace formula for the Laplace-Beltrami operator
{title2=$\Delta f(p)=\sum_i(f\circ\gamma_i)''(0)$}
Choose an <orthonormal basis> $e_i$ of the <tangent space> at $p$ and the unit-speed <geodesics> with initial velocities $e_i$. Their covariant accelerations vanish, so $(f\circ\gamma_i)''(0)=\operatorname{Hess}_g f(e_i,e_i)$. Taking the <metric trace> of the <Riemannian Hessian> proves the formula for $\Delta=\operatorname{div}\nabla$. The <positive Laplace-Beltrami operator> has the opposite sign.
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