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I'm trying to compute the Laplace-Beltrami Operator in local coordinates of some Riemannian manifold $M$. By definition, Laplace-Beltrami Operator

$$\Delta=d\delta+\delta d$$ acts in p-form on $M$, where is defined by $$\delta=(-1)^{n(p+1)+1}\star d\star$$ and $\star$ is the Hodge operator.

So, I started with the $M=\mathbb{R}^n$ and by direct calculation is simple to verifie that, in coordinates $(x_1,\dots,x_n)$ if $\omega=fdx_{i_1}\wedge\dots\wedge dx_{i_p}$ is a p-form on $\mathbb{R}^n$ then

$$\Delta\omega=-\sum_{s=1}^n \frac{\partial^2f}{\partial x_s ^2}dx_{i_1}\wedge\dots\wedge dx_{i_p}.$$

The question is if I have a local coordinates of $\mathbb{S}^n$, for example, $F:\mathbb{S}^n \setminus{(0,\dots,0,1)}\subset\mathbb{R}^{n+1}\to \mathbb{R}^n$ defined by $$F(x_1,\dots,x_{n+1})=\frac{1}{x_{n+1}-1}(x_1,\dots,x_n)$$ how can I write the Laplace-Beltrami operator of the p-form $\omega=fdx_{i_1}\wedge\dots\wedge dx_{i_p}$?

Any idea? Thanks so much.

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    $\begingroup$ I am not sure . But I think you can read the 3.3.46 in 112 page of Jost's Riemannian geometry and geometric analysis. It's $(d^*\alpha)_{i_1...i_p}=-g^{kl}(\frac{ \partial \alpha_{ki_1...i_p} }{\partial x^l} -\Gamma_{kl}^j \alpha _{ji_1...i_p})$. In fact, I still don't know how to get the equation. $\endgroup$
    – Enhao Lan
    Dec 20, 2016 at 11:53

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