On some geometric characteristics of the orbit foliations of the co-adjoint action of some 5-dimensional solvable Lie groups

In this paper, we discribe some geometric charateristics of the so-called MD(5,3C)-foliations and MD(5,4)foliations, i.e., the foliations formed by the generic orbits of co-adjoint action of MD(5,3C)-groups and MD(5,4)-groups.


INTRODUCTION
It is well-known that Lie algebras are interesting objects with many applications not only in mathematics but also in physics. However, the problem of classifying all Lie algebras is still open, up to date. By the Levi-Maltsev Theorem [5] in 1945, it reduces the task of classifying all finite-dimensional Lie algebras to obtaining the classification of solvable Lie algebras.
There are two ways of proceeding in the classification of solvable Lie algebras: by dimension or by structure. It seems to be very difficult to proceed by dimension in the classification of Lie algebras of dimension greater than 6. However, it is possible to proceed by structure, i.e., to classify solvable Lie algebras with a specific given property.
We start with the second way, i.e, the structure approach. More precisely, by Kirillov's Orbit Method [4], we consider Lie algebras whose correponding connected and simply connected Lie groups have co-adjoint orbits (Korbits) which are orbits of dimension zero or maximal dimension. Such Lie algebras and Lie groups are called MD-algebras and MD-groups, respectively, in term of Diep [2]. The problem of classifying general MD-algebras (and corresponding MD-groups) is still open, up to date: they were completely solved just for dimension 5 n  in 2011.
There is a noticeable thing as follows: the family of maximal dimension K-orbits of an MDgroup forms a so-called MD-foliation. The theory of foliations began in Reeb's work [7] in 1952 and came from some surveys about existence of solution of differential equations [6]. Because of its origin, foliations quickly become a very interesting object in modern geometry.
When foliated manifold carries a Riemannian structure, i.e., there exists a Riemannian metric on it, the considered foliation has much more interesting geometric characteristics in which are totally goedesic or Riemannian [8]. Such foliations are the simplest foliations can be on an given Riemannian manifold and have been investigated by many mathematicians. In this paper, we follow that flow to consider some geometric characteristics of foliations formed by K-orbits of indecomposable connected and simply connected MD5-groups whose corresponding MD5-algebras having first derived ideals are 3-dimensional or 4dimensional and commutative. This paper is organized in 5 sections as follows: we introduce considered problem in Sections 1; recall some results about MD(5,3C)algebras and MD(5,4)-algebras in Section 2; Section 3 deals with some results about MD(5,3C)-foliations and MD(5,4)-foliations; Section 4 is devoted to the discussion of some geometric characteristics of MD(5,3C)-foliations and MD(5,4)-foliations; in the last section, we give some conclusions.  Aut G K G  which is defined by K(g)F, X: = F, Ad(g -1 )X for every F * G , X G , gG; where the notation F, X denotes the value of linear form F at left-invariant vector field X. The action K is called co-adjoint representation or K-representation of G in * G and each its orbit is called an K-orbit of S in * G .

Definition 2.3 ([see 2]
). An n -dimensional MD-group or MDn-group is an n-dimensional solvable real Lie group such that its K-orbits in K-representation are orbits of dimension zero or maximal dimension. The Lie algebra of an MDngroup is called MDn-algebra.
Remark 2.4. The family F of maximal dimension K-orbits of G forms a partition of This leads to a foliation as we will see in the next section.
It is well known that all Lie algebras with dimension 3 n  are always MD-algebras. For 4 n  , the problem of classifying MD4-algebras was solved by Vu [10]. Recently, the similar problem for MD5-algebras also has been solved. In this section, we just consider a subclass consists of MD(5,3C)-algebras and MD(5,4)algebras. More specifically, we have the following results.
2)There are 14 families of indecomposable MD(5,4)-algebras which are denoted as follows: Remark 2.7. In view of Proposition 2.6, we obtain 8 families of MD(5,3C)-groups and 14 families of MD(5,4)-groups. All groups of these families are indecomposable, connected and simply connected. For convenience, we will use the same indicates to denote these MD-groups. For example, 5,3,4 G is the connected and simply connected MD(5,3C)-group corresponding to 5,3,4 G .  1) Each geodesic of V that is tangent to L then it lies entirely on L .

 
, V g is called totally geodesic (and TF is called geodesic distribution) if all leaves of F are totally geodesic submanifolds of V . If NF is geodesic distribution, then F is called Riemannian.
Remark 3.4. For any foliation F on (V, g), in the geometric viewpoint, we have 1) F is totally geodesic if each geodesic of V is either tangent to some leaf of F or not tangent to any leaf of F .

CONCLUSION
In this paper, we described some geometric characteristics of subclass of MD5-foliations: the subclass consists of MD(5,3C)-foliations and MD(5,4)-foliations. These results gave concrete examples of the simplest foliations on a special Riemannian manifold (Euclidean space). Recently, a special subclass consists of MD(n,1)algebras and MD(n,n-1)-algebras has been classified for arbitrary n . Therefore, in another paper, we will consider a similar problem for the entire class of MD5-foliations; furthermore, for all MD(n,1)-foliations and MD(n,n-1)-foliations.