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Hi to all!! I have a problem to derive the conformal laplacian
[tex]\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))[/tex]
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
[tex]\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))[/tex]
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