laplacian for metrics on $S^n$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:43:30Z http://mathoverflow.net/feeds/question/70289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70289/laplacian-for-metrics-on-sn laplacian for metrics on $S^n$ unknown (google) 2011-07-14T03:14:13Z 2011-12-15T12:22:12Z <p>It is true that the restiction of the Laplace operator on $\mathbb R^n$ to functions on the sphere is the Laplacian for the round metric on the sphere. Is this true for any Riemannian metric $g$ on $\mathbb R^n$?</p> <p>I mean, is it true that the restriction of $\Delta_g$ to functions on the sphere is the Laplacian on $S^{n-1}$ of the metric induced by $g$?</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/70289/laplacian-for-metrics-on-sn/70296#70296 Answer by Paul for laplacian for metrics on $S^n$ Paul 2011-07-14T05:03:09Z 2011-07-14T05:11:55Z <p>In $\mathbb{R}^n$, in terms of polar coordinates $(r,\theta)$ where $r>0$ and $\theta\in S^{n-1}$, we have the following formula: $$\Delta_{\mathbb{R}^n}=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Delta_{S^{n-1}}.$$ To prove it, you can first try to prove it when $n=2$: When $n=2$, $(x,y)=(r\cos\theta, r\sin\theta)$...I think you can fill out the details.</p> <p>So the answer to your question is yes when $g$ is Euclidean.</p>