I Deriving the conformal Laplacian

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The discussion revolves around deriving the conformal Laplacian, specifically the equation involving the metric g and its relation to the Laplacian and scalar curvature. The context includes the use of a normal frame where all Christoffel symbols are zero at point y. The goal is to express the Laplacian in a conformal manner, referencing Sniatycki's book on geometric quantization. The original poster, Ssnow, later indicates that they have resolved the problem. The exchange highlights the complexities of working with the conformal Laplacian in differential geometry.
Ssnow
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Hi to all!! I have a problem to derive the conformal laplacian

\sum_{m,n}g^{mn}(y)\partial_{q^m}\partial_{q^n}(\psi|\det{(g)}|^{1/4})(y)=\sum_{m,n}|det{g}|^{1/4}(\Delta \Psi -\frac{1}{6}R(y)\psi(y))

where $$g$$ is the metric associated to a Levi Civita connection in a normal frame, we have that in the point $y$ all Christoffel symbols are $$0$$ and $$\Gamma_{ij}^m,k(y)+\Gamma_{ki}^m,j(y)+\Gamma_{jk}^m,i(y)=0.$$

In practice the aim is to find the conformal expression of the Laplacian (Yamabe operator) ... the source is the book "Geometric quantization and quantum mechanics" Sniatycki J.
Thanks,
Ssnow
 
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I solved the problem, Ssnow
 
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