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Given the expression of the Laplace-Beltrami operator $\Delta M$ on a Riemannian manifold $M$ , is there any method for determining the expression of the Laplace-Beltrami operator $\Delta N$ where $N$ is a submanifold of $M$ . Actually I am interested in $N=S(y,r)=(x\in M \mbox{ s.t. } d(x,y)=r)$ where $y\in M$ and $r$ is constant. Thank you Riadh

yess
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Riadh
  • 51

2 Answers2

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In general the $N$ you are interested in will not be a submanifold of $M$ (please search up conjugate points). Nevertheless, as far as I know, they will be submanifolds for almost all $r$. As long as you are within the injectivity radius, you can try using normal coordinates and have an expression of the Laplacian in polar coordinates. That gives you exactly what you are looking for.

MBM
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In general, I don't think that knowing $\Delta$ on $M$ will help; you need to work with the metric on $N$ as usual. Indeed, by Nash's theorem any $M$ can be realized as a submanifold of $\mathbb R^n$ for some $n$. We know that $\Delta u=\sum_{i=1}^n u_{ii}$ on $\mathbb R^n$, but this does not give us any information about $\Delta$ on $N$.

The special case when $N$ is a geodesic sphere may have something in it.