Exponential image and conjugacy classes in the group O(3,2) (Q801423)

Let G be a classical real linear Lie group, g be its Lie algebra and let exp: \(g\to G\) be the exponential map of G. It is now well known the description of conjugacy classes in G and orbits in g under the conjugation action of G. In this paper, the author deals with the determination of conjugacy classes which lie in the interior, boundary or exterior of exp g in G, for the ordinary topology of g and G. For \(G=GL(n,{\mathbb{R}})\) or \(G=SL(n,{\mathbb{R}})\), the author has shown the following [Bull. Fukuoka Univ. Educ., Part III 26, 35-44 (1976; Zbl 0364.22005); ibid. 28, 1-6 (1978; Zbl 0395.20028)]: For an element x of G, (i) x is an interior point of exp g in G if and only if x has no negative eigenvalues, (ii) x is a boundary point of exp g in G if and only if x has negative eigenvalues and their multiplicities are all even. Further, the connected component G of the orthogonal group O(p,q) of the signature (p,q), he showed that exp: \(g\to G\) is surjective if and only if \(q=0,1\) [Mem. Fac. Sci., Kyushu Univ., Ser. A 37, 63-69 (1983; Zbl 0516.22014)]. In this paper, the author deals with groups O(3,2) and O(2,2) and proves the following: (i) an element x of O(2,2) is a boundary point of the image of exp if and only if eigenvalues of x are all real negative and their multiplicities are all even, (ii) an element x of O(3,2) is a boundary point of the image of exp if and only if x is conjugate to \(\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ x'\end{matrix} \right)\) in O(3,2), where x' is a boundary point of the image of exp in O(2,2).