Ends of generic leaves of complete Cartan foliations
Suppose that \(\left( M,\mathcal F\right) \) is a complete Cartan foliation. The author gives the following theorems: If the structure Lie algebra \(\mathfrak{g}_0=\mathfrak{g}_0\left( M,\mathcal F\right)\) is nonzero, then the set of ends of the generic leaves of \(\left( M,\mathcal F\right)\) is either finite and consists of one or two points or is Cantor. If, in addition, \(\dim\mathcal F =2\), then either all generic leaves are noncompact and have genus 0 or all generic leaves are orientable and have only nonplanar ends, or all ends of generic leaves are nonorientable. For \(\mathfrak{g}_0=0\), any finite, countable, or Cantor set can be realized as the set of ends of the generic leaves of \(\left( M, \mathcal F\right)\). If \(\left( M,\mathcal F\right)\) has a compact leaf, then either the set of ends of a generic leaf \(L_0\) is Cantor, or \(e\left( L_0\right)\) is finite and equal to 0,1 or 2; moreover, \(e\left( L_0\right) =0\) if and only if \(\left( M,\mathcal F\right)\) is a compact Riemannian foliation and the space of leaves \(M/\mathcal F\) is a smooth orbifold. Suppose that \(\left( M,\mathcal F\right)\) is of type \(\mathfrak{g}/\mathfrak{h}\) with nonzero structure Lie algebra and at least one of the following two conditions holds: {\parindent7mm \begin{itemize}\item[(1)] the structure algebra \(\mathfrak{g}_0\left( M,\mathcal F\right)\) is solvable; \item[(2)] the algebra \(\mathfrak{g}\) is solvable and the foliation \(\left( M,\mathcal F\right)\) has zero transverse curvature. \end{itemize}} Then the standard generic leaf of this foliation has precisely one or precisely two ends. If \(M\) is a simply connected compact manifold, then \(e\left( L_0\right)\) is equal to 0, 1 or 2. Moreover \(e\left( L_0\right) =0\) if and only if \(\left( M,\mathcal F\right)\) is a compact Riemannian foliation and \(M/\mathcal F\) is a smooth simply connected compact orbifold.
scientific article; zbMATH DE number 5791050