On the Singularity of General Linear Groups

Let G be a Lie group, @ its Lie algebra, then there exists the exponential mapping from @ into G: xexp X, and this mapping is locally homeomorphic at the • zero element O of @.When the exponential mapping:xexp X is not locally homeomorphic at X 0 E @, X 0 is called a singular point of @.And a set {exp tX; t real} is called a path through the unit element E of G.In this paper we shall investigate the path-structure and its singularity of exp@, where exp(~ means the image of the exponential mapping: exp @ = { exp X; Xe@} .Let R and C be the fields of real numbers and complex numbers respectively.In §2, we have a general consideration concerning the singularity of Lie groups, and in § §3 and 4, from our standpoint we shall consider the path-structure and its singularity of the complex general linear group GL(n, C) and the real general linear group GL(n, R) respectively.§ 2. The singularity of Lie groups