A memo on the exponential function and regular points (Q1203536)

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 \(G\).