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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.

We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.

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