Expression for Laplace-Beltrami on sphere?

Question

Is there a good expression for the Laplace-Beltrami on a function $u$ on a sphere or a circle of radius $R>0$ in terms of the Laplacian on ambient space?

There is a formula on Wikipedia for the unit sphere: $$\Delta_{S^{n-1}}f(x) = \Delta f(x/|x|).$$ How about sphere of radius $R$?

Answer 1

$\newcommand{\R}{\mathbf{R}}$To confirm jena's answer: Let $S^{n-1}$ denote the sphere $\sum_{i} x_{i}^{2} = 1$, regarded merely as a smooth manifold. For $R > 0$, let $g(R)$ denote the round metric of radius $R$ on $S^{n-1}$.

The mapping $i_{R}:\bigl(S^{n-1}, g(R)\bigr) \to \R^{n}$ defined by $i_{R}(x) = Rx$ is an isometric embedding, so the image $i_{R}(S^{n-1})$ may be identified with $S^{n-1}(R)$, the sphere of radius $R$ in $\R^{n}$.

Using the argument outlined on Wikipedia, if $f:S^{n-1}(R) \to \R$ is twice continuously-differentiable, define $\hat{f}:\R^{n}\setminus\{0\} \to \R$ by $\hat{f}(x) = f\bigl(Rx/\|x\|\bigr)$, i.e., extend $f$ to be constant along rays through the origin. The induced function $\tilde{f}:S^{n-1} \to \R$ satisfies $$ \tilde{f}(x/R) = \hat{f}(x) = f(x)\quad\text{for $x$ in $S^{n-1}(R)$.} %\tilde{f}(x/\|x\|) = \hat{f}(x) = f(Rx/\|x\|)\quad\text{for $x$ in $\R^{n}$.} $$ In other words, $f$ and $\tilde{f}$ are identical as functions on the smooth manifold $S^{n-1}$, and each is obtained by restricting $\hat{f}$ to the appropriate sphere in $\R^{n}$.

Since $g(R) = R^{2} g(1)$ and the trace of the Hessian depends inversely on the metric components, $$ \frac{1}{R^{2}}\, \Delta_{S^{n-1}(R)} f = \Delta_{S^{n-1}(1)} \tilde{f} = \Delta \hat{f} $$ by the formula for the Laplacian in spherical coordinates.

Answer 2

This is easy to find in the internet. By googling Laplace operator in spherical coordinates, you will find an expression (in the usual form, with three terms) involving partial derivatives with respect to $r$ and the two angle coordinates. Simply remove the term involving differentiation with respect to $r$ (the first one in the usual formatting) and this is what you require. (I would type it here but it is rather complicated). (In the remaining terms, set $r=1$ or $r=R$ for the case of a sphere with radius $R$). The same procedure works for the cicle of radius $R$.