The geometries in N-dimensional Euclidean spaces can be described by Clifford algebras that were introduced as extensions of complex numbers. These applications are due to the fact that the Euclidean invariant (the distance between two points) is the same as the one of Clifford numbers. In this paper we consider the more general extension of complex numbers due to their group properties (hypercomplex systems), and we introduce the N-dimensional geometries associated with these systems. For N > 2 these geometries are different from the N-dimensional Euclidean geometries; then their investigation could open new applications. Moreover for the commutative systems the differential calculus does exist and this property allows one to define the functions of hypercomplex variable that can be used for studying some partial differential equations as well as the non-flat N-dimensional spaces. This last property can be relevant in general relativity and in field theories. © 2005 Birkhäuser Verlag, Basel.

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