CONFORMAL COUPLING AND INVARIANCE IN DIFFERENT DIMENSIONS

Conformal transformations of the following kinds are compared: (1) con formal coordinate transformations, (2) conformal transformations of La grangian models for a D-dimensional geometry, given by a Riemannian ma nifold M with metric g of arbitrary signature, and (3) conformal trans formations of (mini-)superspace geometry. For conformal invariance und er these transformations the following applications are given respecti vely: (1) Natural time gauges for multidimensional geometry, (2) confo rmally equivalent Lagrangian models for geometry coupled to a spatiall y homogeneous scalar field, and (3) the conformal Laplace operator on the n-dimensional manifold M of minisuperspace for multidimensional ge ometry and the Wheeler-de Witt equation. The conformal coupling consta nt xi(c) is critically distinguished among arbitrary couplings xi, for both, the equivalence of Lagrangian models with D-dimensional geometr y and the conformal geometry on n-dimensional minisuperspace. For dime nsion D = 3, 4, 6 or 10, the critical number xi(c) = (4(d-1))/(D-2) is especially simple as a rational fraction.