# Laplacian of the position vector

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.

• 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 $X$. – user10354138 Oct 25 '18 at 16:57