Convex decompositions of real projective surfaces. III: For closed or nonorientable surfaces. (English) Zbl 0958.53022

From the introduction: The purpose of our research is to understand geometric and topological aspects of real projective structures on surfaces. A real projective surface is a differentiable surface with an atlas of charts to \(\mathbb{R} P^2\) such that the transition functions are restrictions of projective automorphisms of \(\mathbb{R} P^2\). Since such an atlas lifts projective geometry on \(\mathbb{R} P^2\) to the surface locally and consistently, one can study the global projective geometry of surfaces.
This paper is the last of a series of papers [J. Differ. Geom. 40, 165-208 (1994; Zbl 0818.53042) and 239-283 (1994; Zbl 0822.53009)]. With this paper, we get a satisfactory classification of all real projective structures on surfaces [see the author and W. Goldman, Bull. Am. Math. Soc., New Ser. 34, 161-171 (1997; Zbl 0866.57001)]. This final paper shows that a nonorientable real projective surface also has an admissible decomposition and a closed real projective surface decomposes into pieces that are convex real projective surfaces or \(\pi\)-Möbius bands.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57M50 General geometric structures on low-dimensional manifolds
53A20 Projective differential geometry