N ov 2 00 3 Essential Constants for Spatially Homogeneous

The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions — some already existing in the literature, and others believed to be new — are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization " general " is given based on the counting of the essential constants, the line-element of each model must contain; indeed, this is the basic contribution of the work. We give two different ways of calculating the number of essential constants for the simply transitive spatially homogeneous (4+1)-dimensional models. The first uses the initial value theorem; the second uses, through Peano's theorem, the so-called time-dependent automorphism inducing diffeomorphisms.