- #1

Sajet

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[itex]-\frac{1}{2} \Delta |\nabla f|^2 = \frac{1}{2} \sum_{i}X_iX_i \langle \nabla f, \nabla f \rangle[/itex]

Where we are in a complete Riemannian manifold, [itex]f \in C^\infty(M)[/itex] at a point [itex]p \in M[/itex], with a local orthonormal frame [itex]X_1, ..., X_n[/itex] such that [itex]\langle X_i, X_j \rangle = \delta_{ij}, D_{X_i}X_j(p) = 0[/itex], and of course

[itex]\langle \nabla f, X \rangle = X(f) = df(X)[/itex]

[itex]\textrm{Hess }f(X, Y) = \langle D_X(\nabla f), Y \rangle[/itex]

[itex]\Delta f = - \textrm{tr(Hess )}f[/itex]

I've tried to use the Levi-Civita identities, but I'm getting entangled in these formulas and don't get anywhere.

Any help is appreciated.