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### Geometrical Constraints for Quantum Mechanics, General Relativity and Cosmology in an N-Dimensional Topology

#### Abstract

Geometrical constraints for general relativity and quantum mechanics are formulated in a multidimensional Cartesian space. A fundamental relationship between these correspondences and complementarity constraints directs us towards a new understanding of the fundamental relationship between relativity and quantum theory. The set of geometrical constraints of this n-dimensional topology are expressed in terms of a hyper-dimensional Minkowski metric, M

_{n}for n > 4 which yields naturally closed cosmological solutions to Einstein’s field equations which also yields compatibility between Einstein’s field equations and the current big bang model without Guth's inflationary model and its possible Googol ~10^{100}solutions as related to string theory. A comprehensive group theoretical approach to the model of the Cartesian space incorporates dark energy and dark matter which results from the model in a natural manner. A lemma is formulated for the relationship of the maximum invariance for an n-dimensional Cartesian space and the dimensions of that space. Group multiplication tables for the M_{n}geometry are formulated for a Cartesian 10 and 11 dimensional spaces.