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Let $M$ be a $n$-dimensional Riemannian manifold (hypersurface) in $\mathbb{R}^{n+1}$, $X=(x_1, ..., x_{n+1})$ is position coordinate vector and $H$ the mean curvature of $M$. I'm having trouble proving the identity
$$\Delta X= (nH)\vec{N}.$$ Where $\nabla$ is the usual Laplace-Beltrami operator on $M$. By definition, if $\{e_i\}_i$ is an orthonormal geodesic frame, $$\Delta X = Trace(Hess(X))=\sum_i\nabla^2 X(e_i, e_i),$$ but am having trouble relating this with the main curvatures in order to get something similar to $H$. I would also appreciate if anyone knows a good introductory reference about this and another Laplacian identities on manifolds.

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  • $\begingroup$ Hint: Recall $\Delta^M$ and $\Delta^{\mathbb{R}^{n+1}}$ differs by the second fundamental form (or the shape operator) term. Choose $e_i$ to be the principal directions at $X$. $\endgroup$ – user10354138 Oct 25 '18 at 16:57

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