Laplace–Beltrami operator as the trace of the Hessian

Let $(M,g)$ be a Riemannian manifold, possibly with boundary, and let $f:M\to \mathbb{R}$ be smooth. At each interior point, the Laplace–Beltrami operator of $g$ is defined by the metric trace of the Hessian, ${\mathrm{\Delta }}_{g}f(p):={\mathrm{tr}}_g(\mathrm{Hess}f{)}_p:={\sum }_{i=1}^n\mathrm{Hess}f(e_i,e_i),$ where $n=\dim M$ and $e_1,\dots ,e_n$ is any $g_p$-orthonormal basis of $T_{p}M$; the trace of a bilinear form with respect to a positive-definite inner product does not depend on that choice, so ${\mathrm{\Delta }}_{g}f$ is well-defined and smooth in the interior; when $f$ is smooth up to the boundary, it has the usual one-sided extension there. Here $\mathrm{Hess}f(X,Y)=g({\mathrm{\nabla }}_X\mathrm{grad}f,Y)$ is the Levi-Civita Hessian of Gradient hessian and divergence connection formulas.

Equivalently, and with the positive-divergence convention of this library, ${\mathrm{\Delta }}_{g}f={\mathrm{div}}_g(\mathrm{grad}f),$ where ${\mathrm{div}}_{g}X=\mathrm{tr}(v\mapsto {\mathrm{\nabla }}_{v}X)$ is the Riemannian divergence of Riemannian divergence as computed in Gradient hessian and divergence connection formulas. The equivalence is the pointwise computation ${\sum }_i\mathrm{Hess}f(e_i,e_i)={\sum }_{i}g({\mathrm{\nabla }}_{e_i}\mathrm{grad}f,e_i)=\mathrm{tr}(v\mapsto {\mathrm{\nabla }}_v\mathrm{grad}f),$ the middle sum being the endomorphism trace of $v\mapsto {\mathrm{\nabla }}_v\mathrm{grad}f$ in the orthonormal basis $(e_i)$, which the same supplier identifies with ${\mathrm{div}}_g(\mathrm{grad}f)$; no local coordinates are needed and no analytic regularity beyond the smoothness of $f$ is used. On a zero-dimensional manifold both sides are the empty sum $0$, and on a manifold with boundary the operator is defined at interior points, with the usual one-sided extension at boundary points when $f$ is smooth up to the boundary.

This definition is the pointwise geometric operator only. It records the trace formula, the divergence form and the sign convention ${\mathrm{\Delta }}_g={\mathrm{div}}_g\circ \mathrm{grad}$ fixed by Riemannian divergence. It does not assert any analytic conclusion about harmonic functions on manifolds — no maximum principle, no Harnack inequality, no boundary-value solvability, no spectral theory, and no Liouville theorem. Those are separate results with their own hypotheses, and their Euclidean forms are not licensed on a general Riemannian manifold merely by writing down this definition.