Lie Groups Actions on Non Orientable &amp;lt;i&amp;gt;n&amp;lt;/i&amp;gt;-Dimensional Complex Manifolds

T Cao-Huu,D Ghisa

doi:10.4236/apm.2021.116039

Abstract

Analytic atlases on 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 with the two elements group of a fixed point free antianalytic involution of . Involutions h(z) of this kind are obtained linearly by composing special Möbius transformations of the planes with the mapping . A convenient partition of is performed which helps defining an internal operation on and finally actions of the previously defined Lie groups on the non orientable manifold are displayed.

Highlights

We dealt in [1] and [2] with Lie groups of bi-Möbius transformations defined on
We have introduced in previous works actions of Lie groups on non orientable Klein surfaces
The mapping α : G × n h → n h defined by α = z w is a left action of the Lie group G on the non orientable manifold n h

Summary

Introduction

We dealt in [1] and [2] with Lie groups of bi-Möbius transformations defined on. The concept can be extended linearly to n in the following way. F (z, w) = a only if z = a or w = a and f (z, w) = a−1 only if z = a−1 or w = a−1 It results that the composition law z w = f (z, w) defines a structure of Abelian group on n with the unit element 1 and z−1 the inverse element of z. Due to the fact that gzk are injective and taking into account Theorem 1 (6), gzk ( wk ) = ak if and only if wk = ak and gzk ( wk ) = 1 ak if and only if wk = 1 ak These Möbius transformations induce transformations of n defined by ( ) gz (w) = gz ( w1 ), gz ( w2 ), , gzn ( wn ). Theorem 1 implies that G acts freely and transitively on itself by left and right translations

Discrete Subgroups of G

Antianalytic Involutions of n

Conclusion