Hofmann, Karl H. A memo on the exponential function and regular points. (English) Zbl 0768.22002 Arch. Math. 59, No. 1, 24-37 (1992). Let \(G\) denote a connected real Lie group with Lie algebra \(\mathfrak g\) and \(U = \text{Reg}(G)\) the open dense subset of regular points. Theorem. (i) \(U \cap \exp {\mathfrak g}\) is closed in \(U\). (ii) \(\overline{\exp{\mathfrak g}} \subseteq (\exp {\mathfrak g}) \cup (G\setminus U)\). (iii) \(U \cap \exp {\mathfrak g} = \exp(\text{Reg}({\mathfrak g}) \cap \text{reg}\exp)\), where \(\text{Reg}({\mathfrak g})\) is the set of regular points of the Lie algebra \(\mathfrak g\) and \(\text{reg}\exp\) is the set of points of \(\mathfrak g\) at which the exponential function is nonsingular. – We conclude that the exponential function has dense image in \(G\) if and only if it contains \(U\). In the process of proving these results we prove a foliation theorem for \(U\) which allows us a good understanding of the function which associates with an element \(g\in U\) the Cartan algebra \({\mathfrak h}(g)\), the nilspace of \(\text{Ad}(g)-1\). We operate in the Grassmann manifold containing the Cartan algebra of \(\mathfrak g\) and the compact space of all closed subgroups of \(G\). Reviewer: K.H.Hofmann (Darmstadt) Cited in 7 Documents MSC: 22E15 General properties and structure of real Lie groups 22E60 Lie algebras of Lie groups 53C12 Foliations (differential geometric aspects) 22E25 Nilpotent and solvable Lie groups Keywords:connected real Lie group; Lie algebra; regular points; exponential function; foliation; Cartan algebra; Grassmann manifold PDFBibTeX XMLCite \textit{K. H. Hofmann}, Arch. Math. 59, No. 1, 24--37 (1992; Zbl 0768.22002) Full Text: DOI References: [1] N.Bourbaki, Intégration, Chapitres 7 et 8. Paris 1963. [2] N.Bourbaki, Groupes et algébres de Lie, Chapitres 2 et 3. Paris 1972. [3] N.Bourbaki, Groupes et algébres de Lie, Chapitres 7 et 8. Paris 1975. [4] G.Gierz, K. H.Hofmann, J. D.Lawson, K.Keimel, M.Mislove and D. S.Scott, A Compendium of Continuous Lattices. Berlin-Heidelberg-New York 1980. · Zbl 0452.06001 [5] K. H.Hofmann, A memo on the singularities of the exponential function. Seminar Notes 1990. [6] K. H. Hofmann andA. Mukherjea, On the density of the image of the exponential function. Math. Ann.234, 263-273 (1978). · Zbl 0382.22005 · doi:10.1007/BF01420648 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.