1. Introduction
The Riemanean geometry is the study of curved surfaces. One might have a cylinder, or a sphere. For instance, one can use a cardboard paper towel roll to study a cylinder and a globe to study a sphere. In such spaces, the shortest curve between any pair of mass points on such curved surface is called a minimal geodesic or geodesic. One can find a geodesic between two mass points by stretching a rubber band between them. The first thing that will be noticed is that sometimes there is more than one minimal geodesic between two mass points. For instance, there are many minimal geodesics between the north and south poles of a sphere. This leads to derive more generalized formulas, which are true in the Euclidean space. Therefore, one of our main concern in this work is to motivate researcher for finding exact formulas for computing Barycenters on non-Euclidean spaces, namely on manifolds. In [1] , the authors suggest an estimation of the so-called Riemannean Barycenters. For a use in the same context, we refer for instance to [2] . Where, in [3] Riemannian means were presented as solution of variational problems. We think that for special case, for instance smooth manifolds, we could derive similar formulas to the Euclidean case, define and study interesting properties of Riemannean Barycenters.
In this work, we begin by setting out some basic theory of the geodesic of the unit circle as one dimensional manifold, we define the multiple Barycenters and compute it for some special cases. For more details, we refer to [4] -[6] .
Let I be an interval of real numbers. Then, a curve γ is a continuous mapping given as 
where 
is a smooth Riemannean manifold. The curve γ is said to be simple if it is injective. If I is a closed bounded interval [a, b], we also allow the possibility 
(this convention makes it possible to talk about closed simple curve). If 
for some 
(other than the extremities of I), then 
is called a double (or multiple) point of the curve. A curve γ is said to be closed or a loop if I = [a, b] and if
. A closed curve is thus a continuous mapping of the circle. Let x, y are two points of 
subset of
. If γ is a curve such that
, 
and 
we said that γ is a connection between x and y. We define 
the set of all connections between x and y on
. Let x, y be two mass points of
. The geodesic is the minimum distance between x and y on 
(or the shortest path), denoted by
, which is given by
. We say that X is a geodesic space if for any pair 
there is a minimal geodesic from x to y and this is given as a minimal of
, where 
is a curve joining x and y. Let two points
. The geodesic (shortest path) between the two points x and y is given by
, where
, is the scalar product on
. The shortest geodesic (which is a metric on 
that joins two non antipodal points. The geodesic distance between the two points is precisely equal to α. A midpoint m of two masses on 
is defined 
where g is a geodesic on
. For more details, we refer to [7] -[9] .
Example 1. Consider the following examples on the unit circle:
1) Each point on 
is midpoint of itself 
2) The midpoint of 
and 
is the unique point
, therefore
3) Observe that even if 
for
. The fact that 
is not a geodesic between x and y.
4) The midpoint on 
is not unique: Consider 
(north pole) and 
(south pole),
.
2. Barycenters on 
The arithmetic mean of a finite set of mass points 
the Euclidean space
. is the unique point x that minimizes the sum of the squared distances to the given mass point, that is
, where d(·,·)
is an Euclidean distance in
. The point realizing the minimum of the function 
is called also Barycenter of the set of mass points
. In this section we would like to extend this definition to a smooth Riemannean manifold
, and for more information, we refer to [1] [3] .
Definition 1. The Barycenter of the set of mass points 
subset 
with weighs 
(respectively) is the mass point minimizing the following functional (Energy function):
where g(·,·) is the geodesic in 
and μ is the distribution in the continuous case.
Lemma 1. Let 
and 
are two mass points with weights 
and
, such that 
with
, 
and
. If 
is the geodesic Barycenter of 
in
, then there exists 
such that 
Proof. We prove that the geodesic Barycenter of two mass points is a mass point contained on the shortest path. We have
Suppose that 
is not element of
. There exists 
such that
it follows
(
is not element of
). It implies
In fact, the point 
satisfies the minimal and the following equality 
Lemma 2. Let two mass points 
with weights 
and 
respectively. If for all 
are unique, then the geodesic Barycenter of 
is unique.
Proof. Suppose that the Barycenter is not unique, and denote by 
and 
solving the energy functional. From Lemma 1, we know that the geodesic Barycenter is contained on the shortest path between
, it means
there exists two geodesics connecting
, this contradicts the uniqueness of the geodesic.
Definition 2. Let an oriented unit circle with origin
, and a subset of mass points points
. We denote by 
the number of all geodesics, their corresponding curves contain the origin 
where 
3. Geodesic Multiple Barycenters on 
The existence of Barycenters on 
interpreted as one dimensional Riemannean space does assure it uniqueness. In this section, we give a general condition of the existence of a geodesic multiple Barycenter on
.
Definition 3. (Geodesic Multiple Barycenter). We said that a set of mass points have a multiple Barycenter if the solution of the energy function is not unique.
Example 2.
1) The Barycenter of the north and south poles is not unique i.e. 0 and
.
2) We examine a simple example on the unit circle. Compute the Barycenter of four mass points
Define 
It implies
Differentiations with respect to z
1) 
with
:
From 
it follows 
and
.
2) 
with
:
From 
it follows 
and
.
3) 
with
:
From 
it follows 
and
.
4) 
with
:
From 
it follows 
and
.
We conclude that 
4 are Barycenters of the set X.
Definition 4. (Symmetry propriety SP). Let 
be a mass distribution on the unit circle. The map 
is called a symmetry of 
if: σ is an isometry and 
Example 3.
, for which 
satisfies the map
the symmetry propriety.
The mass distribution is defined on two points
, 
on 
such that 
It is clear that for
,
. If the distribution is defined over the set of equidistant masses 
then for 
with 
-rotation on the unit circle, with a supplementary propriety that 
such that 
Figure 1 shows some example of symmetry mass points on 
(From left to right) On the first figure, we have two mass points antipodal, the first on (1, 0) Cartesian coordinate and the second in (0, 1). The second, third and fourth show other types of semmetries.
Theorem 1 (Multiple Barycenter). Let 
on 
be a set of n mass points. If there exists a symmetry map Aθ, then x admits a multiple Barycenter.
Proof. For n = 2 and 
is the symmetry map such that
. Then for all
hold,
It follows from our assumptions that 
for all 
It is clear that 
for all
and if for all 
is a Barycenter of for all 
then 
is too. In this case we have double Barycenters. Now suppose that n > 2. Then, for all 
we have
Moreover
Thus
Lemma 3. Under the same assumptions of Theorem 1. Let k be the number of the symmetry maps 
in S1 which have the propriety
, i.e. the distribution of the mass points is 
-invariant. The number of the multiple Barycenters is k + 1.
Proof. The proof is left to the reader.
Figure 2 contains three, four, five, six, eight and nine mass points satisfying the symmetry property and non necessary with same weights. The geodesic between two arbitrary and successive mass points is constant. I each of the figures our code generates all possible Barycentes so-called multiple Barycenters.
Figure 3 shows the uniqueness of the Barycenter on 
in the case of four masses without symmetry property.
Figure 1. Examples of symmetry masses distribution on S1.
Figure 3. Case of non-symmetry on S1 (uniqueness of the Barycenter. (small perturbation of the north and south poles respectively.
4. Concluding Remarks
As consequence of the non-uniqueness geodesic between two arbitrary mass points, we have presented an example on the notion of multiple Barycenters. Moreover, we have derived some properties similarly to the Euclidean space. We believe that more interesting cases are to construct significant formulas for arbitrary manifolds.
Acknowledgements
The authors would like to dedicate this work to Dr. Musaed Al-Subhi Rahimaho Allah.
NOTES
*Corresponding author.