Limit sets of projectively flat manifolds (Q2706531)

Let \( M \) be a projectively flat manifold. Fix a developing mapping \( D: \widetilde{M} \rightarrow \mathbb{R}P ^n \) and a Riemannian metric on \( \mathbb{R}P ^n \). Let \( L _\infty (M) \) be the image of the ideal boundary of \( \widetilde{M} \) under the unique extension \( \overline{D}: \overline{M} \rightarrow \mathbb{R}P ^n \) of \( D \), where \( \overline{M} \) denotes the metric completion of \( \widetilde{M} \) [\textit{S. Choi}, J. Differ. Geom. 40, 165-208 (1994; Zbl 0818.53042)], \( L _E (M) \) the set of those points \( y \) which is the end point \( c(1) \) of a continuous curve \( c \) in \( \mathbb{R}P ^n \) such that there exists a curve \( \widetilde{c}(t) \in \widetilde{M}\) \((0 \leq t < 1)\), \(D(\widetilde{c}(t))=c(t) \) and \( \widetilde{c}(1) \) cannot be defined continuously in \( \widetilde{M} \) [\textit{N. H. Kuiper}, Ann. Math. 50, 916-924 (1949; Zbl 0041.09303)], \( L _O (M) \) the set of points \( y \) such that the inverse image of any compact neighborhood of \( y \) under \( D \) has a nonempty and noncompact component [\textit{R. S. Kulkarni} and \textit{U. Pinkall}, Lect. Notes Math. 1209, 190-209 (1986; Zbl 0612.57017), \textit{S. Matsumoto}, ``Foundations of flat conformal structure'' (preprint)], \( L _K (M) = K(\Gamma) \cap \overline{\Omega} \), where \( \Gamma \subset \text{PGL}(n+1, \mathbb{R}) \) is a holonomy group of \( M \), \(K(\Gamma) \) the union of kernels of all singular projective transformations in the closure of \( \Gamma \) in \( \text{P}\mathfrak{gl} (n+1, \mathbb{R}) \) [= the projectivization of the vector space \( \mathfrak{gl} (n+1, \mathbb{R}) \)] and \( \Omega = D(\widetilde{M}) \), and \( L _J (M) = L _J (\Gamma) \cap \overline{\Omega} \), where \( L _J (\Gamma) \) is the set of those points where \( \Gamma \) is not equicontinuous. If \( M \) is closed, the author proves that \( L _\infty (M) = L _E (M) \subset L _O (M) \subset L _K (M) = L _J (M) \). Some examples, showing that in general \( L _O (M) \subset L _K (M) \) is a strict inclusion, are considered.