Kaluza-klein Reduction of Conformally Flat Spaces

A “conformal tensor” is constructed from the metric tensor gMN (or Vielbein e A M ) and is invariant against Weyl rescaling gMN → egMN (or eM → eeM ). Moreover, it vanishes if and only if the space is conformally flat, gMN = e ηMN (or e A M = eδ M ). In dimension four or greater the conformal tensor is the Weyl tensor. In three dimensions the Weyl tensor vanishes identically, while the Cotton tensor takes the role of probing conformal flatness and Weyl invariance. Of interest is the behavior of the n-dimensional conformal tensor under a KaluzaKlein dimensional reduction to n − 1 dimensions. Specifically in n dimensions we take the metric tensor in the form