Lie Groups Actions on Non Orientable <i>n</i>-Dimensional Complex Manifolds

Abstract

Analytic atlases on n can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-Möbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing n  with the two elements group of a fixed point free antianalytic involution of n  . Involutions ( ) h z of this kind are obtained linearly by composing special Möbius transformations of the planes with the mapping ( ) 1. A convenient partition of n  is performed which helps defining an internal operation on n  h and finally actions of the previously defined Lie groups on the non orientable manifold n  h are displayed.

Multipliers and Classification of <i>m</i>-Möbius Transformations

“…The m-Möbius transformations have been introduced in connection with Lie groups' actions on complex manifolds (see [3] and [4]). They represent an interesting mathematical topic in itself and we dedicated ourselves to performing in this paper a study of these transformations parallel to that of classical Möbius transformations of the complex plane.…”