Conformal Coupling and Invariance in Different Dimensions

Conformal transformations of the following kinds are compared: (1) conformal coordinate transformations, (2) conformal transformations of Lagrangian models for a D-dimensional geometry, given by a Riemannian manifold M with metric g of arbitrary signature, and (3) conformal transformations of (mini-)superspace geometry. For conformal invariance under this transformations the following applications are given respectively: (1) Natural time gauges for multidimensional geometry, (2) conformally equivalent Lagrangian models for geometry coupled to a spacially homogeneous scalar field, and (3) the conformal Laplace operator on the n-dimensional manifold M of minisuperspace for multidimensional geometry and the Wheeler de Witt equation. The conformal coupling constant ξc is critically distinguished among arbitrary couplings ξ, for both, the equivalence of Lagrangian models with D-dimensional geometry and the conformal geometry on n-dimensional minisuperspace. For dimension D = 3, 4, 6 or 10, the critical number ξc = D−2 4(D−1) is especially simple as a rational fraction. PACS No.: 0460, 0240