General Relativity and Gravitation, Vol. 31, No. 5, 1999 Re¯ections on the Application of the Geometric Approach to Quantum Mechanics to General Relativity Alejandro Corichi 1,2 and Michael P. Ryan, Jr. 1 Received October 26, 1998 This article is a series of remarks on the application of the geometric ap- proach to quantum mechanics to gravitation. Bianchi Type I cosmologies are used as minisuperspace models in order to give concrete examples of the problems one expects to encounter. This article is meant as a preliminary look at the problem of the applica- tion of a relatively new approach to quantum theory to the gravitational ®eld. This approach considers the geometrical structure of quantum state spaces, endowing them with a KÈ ahler metric which gives a sympletic form and a Riemannian metric which determine both the evolution of states (U- processes) and the probability structure of measurements (R-processes). The approach as been applied to a number of quantum systems, in the main spin systems, where the state space is ®nite-dimensional, in order to avoid problems associated with the more common in®nite-dimensional state spaces and the even more complicated state spaces associated with ®eld theories. Here we plan to make some remarks, without attempting a complete analysis, on the quantization of the gravitational ®eld in the context of simpler constrained systems such as the the one-dimensional harmonic oscillator with parametrized time and Bianchi Type I quan- tum cosmologies, that, while they have a ®nite-dimensional classical state 1 Instituto de Ciencias Nucleares, Universidad Nacional Aut  onoma de M exico, A. Postal 70-543, M exico D.F. 04510, M exico 2 E-mail: corichi@nuclecu.unam.mx 621 0001-7701/ 99/ 0500-0621$16.00/ 0 ° c 1999 Plenum Publishing Corporation