1. Introduction
We dealt in [1] and [2] with Lie groups of bi-Möbius transformations defined on
. The concept can be extended linearly to
in the following way.
If
then
,
,
. Let us study the function
defined by
(1)
where
,
,
,
,
,
.
Theorem 1. Let us denote
and
,
. The function
satisfies the following relations:
1)
, for every
2)
and
, for every
3)
, for every
4)
, hence
, for every
5)
for every
6)
and
for every
. Moreover,
only if
or
and
only if
or
.
It results that the composition law
defines a structure of Abelian group on
with the unit element
and
the inverse element of
.
Proof. The proof requires only elementary computation. For (5) it is enough to show that
for arbitrary
and this results again after an elementary (although a little more tedious) computation. The relation (6) shows that by removing the elements
and
we obtain a subgroup
of this group. The functions
(2)
are Möbius transformations, since
as long as
and
, which has been postulated. Moreover, due to the fact that
are injective and taking into account Theorem 1 (6),
if and only if
and
if and only if
. These Möbius transformations induce transformations of
defined by
(3)

Theorem 2. The set of transformations
endowed with the composition law
(4)
is an Abelian group having the identity element
and such that the inverse element of
is
.
This group is isomorphic with
, the isomorphism being given by the mapping
. It makes
a Lie group with analytic structure as n-dimensional complex differentiable manifold.
Proof. Indeed, if
, then by Theorem 1 (6)
, hence
.
The commutativity results from:
.
The identity element is
, since
.
The composition law is associative since:
.
Finally, the inverse element of
is
since
for every
.
It is obvious that the mapping
is bijective. Since
are analytic functions in
, the function
is analytic in
. Obviously,
as complex n-dimensional manifold has an analytic structure and then the isomorphism
makes from
a Lie group with analytic structure as complex n-dimensional manifold. 
We used [3] [4] [5] for the basic knowledge about Lie groups and their actions.
The actions by left and right translations of
on itself are defined as:
, respectively
.
Theorem 1 implies that
acts freely and transitively on itself by left and right translations.
2. Discrete Subgroups of G
Let
be an arbitrary element and for every
let us denote
(5)
where
and
.
Then, for every
we have
and using the formula (4),
for every
, in particular
which is the identity element in
. It results that the group
generated by
is a subgroup of
. By Theorem 1 (2) we have that
only if
, hence for every
we have
and since for every
,
implies
then for
and
, we have
, therefore the elements of
are all distinct.
Theorem 3. For every
, the group
generated by
is a discrete subgroup of
.
Proof. The case of
is trivial. Suppose that for a given
we would have
then
, which means that
, contrary to the assumption. 
Corollary 1. For every
the subgroup
acts freely and properly discontinuously on
by left and by right translations.
3. Antianalytic Involutions of
Let
be a non-empty subsequence of
and let
(6)
Then the mapping
defined by
(7)
is a fixed point free involution of
in the sense that some of the mappings
are fixed point free involutions of
, while the others are the identity mapping. Then, it is true for
itself that for every
we have
. Moreover, there is no
for which
, since this would imply
for every j and if
this would mean
, which is absurd. Since
are antianalytic self mappings of
, we will say that
is antianalytic.
Let us notice that the functions
of the form we just listed are not the only antianalytic involutions of
. If for
,
and
we take the Möbius transformation
which maps the unit disc onto itself, the unit circle onto itself and the exterior of the unit disc onto itself, we can prove:
Theorem 4. The function
, where
, is a fixed point free antianalytic involution of
.
Proof. We have that
(8)
(9)
(10)
which shows that
is antianalytic, since its complex conjugate is analytic. The equality
implies
, which is impossible since h is fixed point free, hence
is fixed point free. Finally,
(11)
showing that
is an involution.
We keep the notation
for any antianalytic involution of
constructed with the functions of this type by the method of the previous paragraph.
A given antianalytic involution
and the identity mapping of
form a group of transformations
of
. 
Theorem 5. The quotient space
can be endowed with a differentiable manifold structure, so that it becomes a non orientable differentiable manifold.
Proof. Indeed, an analytic atlas can be created on
such that the local chart for any point
is the identity on
. Next, if for a
we have
when
local charts
can be used, where
with
for
and
otherwise and
when
and
otherwise. Obviously,
with such an atlas is a differentiable manifold for which every change of chart is a complex analytic function. The projection function
, by which we have
for every
, induces a differentiable manifold structure on
. Indeed, to every chart
on
corresponds a chart
on
, where
and if
then
, hence
. This structure is no more analytic since every change of charts
is antianalytic. However, it is harmonic and therefore of class
. 
Theorem 6. For any fixed point free antianalytic involution
of
there is a partition of
into two sets
and
such that
if and only if
. With the induced topology of
the topological spaces
and
, as well as
are homeomorphic under the projection
.
Proof. We give a constructive proof. Let
be arbitrary. Since
is a fixed point free involution of
we have that
. Then there are disjoint open neighborhoods
of
and
of
such that
if and only if
. Let
be arbitrary. We infer that
. Indeed, supposing
would imply that
, contrary to the hypothesis. Analogously we find a contradiction supposing
. Then there are open disjoint neighborhoods
of
and
of
such that
if and only if
. Moreover, we can take
such that
. In this way we can build two sequences of open sets
and
such that
and
are disjoint and
if and only if
. To make sure that the process ends in a countable number of steps, we can decide to take all the points
such that their coordinates in
are rational. Moreover,
and
are then maximal in the sense that
does not contain any open set. The set
with the trace atlas of
is a complex differentiable manifold of dimension less than n. We repeat the process for
and we find that there are two relatively open maximal sets
and
such that
if and only if
. Moreover,
is a complex differentiable manifold of dimension less than that of
and the process continues
steps until we obtain dimension 0. Then
is a countable set such that
if and only if
. Then a partition of
into
and
such that
if and only if
is straightforward. Let us denote
and
. Then
and
are such that
,
and
if and only if
.
It is obvious that
and
are one-to-one and onto functions and if the topology of
is chosen such that the projection
is continuous, then the three topological spaces are homeomorphic.
We needed this construction for the following reason. The notation
is ambiguous in the sense that on the right hand side we have an ordered couple of points, while in reality for the
the order does not count and there is a situation which will appear later where this fact is essential. Now we can decide that once
and
have been built, they will remain permanently the same and every time we meet a couple
we have chosen
. It is as if we ignore occasionally the existence of
and instead of
we work only with
.
We can define an operation on
by using the composition law in
from the section 1. For every couple
and
from
we write
, where
. 
Theorem 7. The multiplication
is an internal composition law in
with the unit element
and
, the inverse element of
. The multiplication is commutative but not associative and therefore this law does not define a structure of Lie group on
.
Proof. It is obvious that for every
,
is well defined and represents an element of
. Moreover,
,
and
. The non associativity of the law comes from the fact that although
it may happen that
and then the expression
has no meaning. 
We were expecting that one of the group axioms of the multiplication in
not to be satisfied, since otherwise this manifold would be a Lie group and it is known (see [5], page 140) that every Lie group is an orientable manifold. However, as proven in the next theorem, actions of Lie groups on such a manifold exist.
Theorem 8. The mapping
defined by
is a left action of the Lie group
on the non orientable manifold
.
Proof. The mapping is obviously of class
. Moreover,
for every
. Finally,
, hence indeed,
is a left action of the Lie group
on the non orientable manifold
. 
Corollary 2. The mapping
defined by
is a right action of
on
, since
, as it can be easily checked.
By the general theory of Lie groups, these actions of
on
define homomorphisms t from the Lie group
to the group
of diffeomorphisms of
such that
is the mapping
, respectively
. Reciprocally, every homomorphism
defines left and right actions of
on
by
, respectively
.
4. Conclusion
Non orientable n-dimensionl complex manifolds can be obtained by factorization with a two elements group generated by an antianalytic involution of
. Such involutions can be obtained, for example, composing in some coordinate planes Möbius transformations of the form
,
with the mappings
. An internal composition law can be defined on such a manifold with the help of some bi-Möbius transformations and actions of Lie groups on the respective manifold can be put into evidence. We realized this task by devising an appropriate partition of that manifold.