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Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M f$ in terms of $\Delta^{S}f$ ? ((i.e the relation between them))

PS: $\Delta^M$ and $\Delta^{S}$ represent the Laplace-Beltrami operator the sub-manifold $M$ and the manifold S respectively.

Thank you

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Let $e_1,\cdots e_m$ be local orthonormal basis on $M$ and $e_{m+1} , \cdots e_s$ be a local orthonormal frame normal to $M$. Let $\overline \nabla$ and $\nabla$ be the covariant derivative on $S$ and $M$ respectively.

$$\begin{split} \Delta^S f &= \sum_{k=1}^s \overline\nabla^2 f(e_k, e_k) \\ &= \sum_{k=1}^m \overline\nabla^2 f(e_k,e_k) + \sum_{k=m+1}^s \overline \nabla ^2 f(e_k,e_k) \end{split}$$

The first term on the right can be written as

$$\begin{split} \sum_{k=1}^m \overline \nabla^2 f(e_k,e_k) &= \sum_{k=1}^m e_k e_k f - \overline\nabla_{e_k} e_k f\\ &= \sum_{k=1}^m e_ke_k f - \nabla_{e_k } e_k f - (\overline\nabla_{e_k}e_k)^\perp f \\ &= \Delta^M (f|_M) - \vec H f, \end{split}$$

where $\vec H$ is the mean curvature vector of $M$ in $S$. So

$$(\Delta^S f)|_M = \Delta^M f|_M - \vec H f + \sum_{k=m+1}^s \overline \nabla ^2 f(e_k,e_k) .$$

It seems that not much can be said about the last term (which cannot be seem from $f|_M$).

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  • $\begingroup$ I think that you have some mistake, because I have got $H\frac{\partial f}{\partial\nu}$ in the intermediate align instead of $Hf$ @JohnMa $\endgroup$ – A s May 29 '16 at 10:24
  • $\begingroup$ @As What's your $\nu$? $\endgroup$ – user99914 May 29 '16 at 10:29
  • $\begingroup$ $\nu$ is unit outward normal of $M$ in $S$ $\endgroup$ – A s May 29 '16 at 10:30
  • $\begingroup$ But if $M$ is not of codimension one, there is no unit outward normal. @As $\endgroup$ – user99914 May 29 '16 at 10:31
  • $\begingroup$ No, because of $M$ is of co-dimension one, this means that $M$ is nothing but $\partial S$, and in that case $\nu$ is the outward 'unit' normal of $\partial S$ $\endgroup$ – A s May 29 '16 at 10:33
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If you are using $$\Delta f = \frac{1}{\sqrt{detg}}d _i (g^{ij}d_jf)$$ formula , where summation is occurring. Then, by picking a local orthonormal frame on $M$, and then extending to an orthonormal frame on $S$, I think you can simply ignore those derivatives in the above formula that involve directions other than those on $S$.

Since we are dealing with an intrinsic construction, this should work. Check with $R^2$ and $R^1$ case, for instance.

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