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For a manifold of dimension $n+1$, consider a metric $\tilde{g}=s^{2}\left(g-d s^{2} / s^{2}\right)$, where $g$ is the metric of a co-dimension $1$ submanifold. I calculate its laplacian to be $$\tilde{\Delta}=s^{-2}\left[\Delta_{g}-\left(s \partial_{s}\right)^{2}-n s \partial_{s}\right]$$ However, this paper on pg 23 writes it as $$\tilde{\Delta}=s^{-2}\left[\Delta_{g}+\left(s \partial_{s}\right)^{2}+n s \partial_{s}\right]$$ Because a lot of further calculations assume this formula, I cannot ignore it or think that it is a typo. Could someone tell me if they are getting the same formula as me?

I take the formula of the laplacian to be the trace of the hessian, and not its negative, as it is sometimes used in other conventions (although both conventions give the wrong answer here).

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  • $\begingroup$ I did a quick computation and arrive at a solution similar to your's. However, if you compute this via a conformal change of the metric shouldn't there be a factor $n-1$ instead of $n$ for a $n+1$-dimensional manifold? $\endgroup$
    – AlexD
    Jan 2, 2022 at 12:30

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