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

Hoa Quang Duong,Tuan Anh Nguyen,Vu Anh Le

doi:10.32508/stdj.v18i2.1081

Abstract

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.

Highlights

It is well-known that Lie algebras are interesting objects with many applications in mathematics and in physics
By the LeviMaltsev Theorem [5] in 1945, it reduces the task of classifying all finite-dimensional Lie algebras to obtaining the classification of solvable Lie algebras
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

Summary

INTRODUCTION

It is well-known that Lie algebras are interesting objects with many applications in mathematics and in physics. By Kirillov's Orbit Method [4], we consider Lie algebras whose correponding connected and connected Lie groups have co-adjoint orbits (Korbits) which are orbits of dimension zero or maximal dimension. There is a noticeable thing as follows: the family of maximal dimension K-orbits of an MDgroup forms a so-called MD-foliation. 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]. We follow that flow to consider some geometric characteristics of foliations formed by K-orbits of indecomposable connected and connected MD5-groups whose corresponding MD5-algebras having first derived ideals are 3-dimensional or 4dimensional and commutative. 2) There exist exactly 3 topological types F 3 , F 4 , F 5 of 14 families of considered MD(5,4)-foliations as follows:. We describe some geometric characteristics of considered MD(5,3C)-foliations and MD(5,4)-foliations

Foliations of the type F 1

Foliations of the type F 2

CONCLUSION