Convex real projective structures on compact surfaces. (English) Zbl 0711.53033

The purpose of this paper is to investigate the space P(s) of inequivalent convex real projective structures on a compact surface S with \(\chi (S)<0\), i.e. a structure on S such that its universal covering may be identified with a convex domain \(\Omega \subset {\mathbb{R}}{\mathbb{P}}^ 2\) and its fundamental group acts as a discrete group \(\Gamma\subset PGL(3,{\mathbb{R}})\) of projective transformation (acting properly on \(\Omega\)): \(S=\Omega /\Gamma\). An equivalence for such structures on S is defined up to a projective image h(\(\Omega\)) and a conjugation \(h\Gamma h^{-1}\) where \(h\in PGL(3,{\mathbb{R}}).\)
The author’s main result is: Let S be a compact surface having n boundary components such that \(\chi (S)<0\). Then P(s) is diffeomorphic to a cell of dimension -8\(\chi\) (s) and the map which associates to a convex \({\mathbb{R}}{\mathbb{P}}^ 2\)-manifold M the germ of the \({\mathbb{R}}{\mathbb{P}}^ 2\)- structure near \(\partial M\) is a fibration of P(s) over an open 2n-cell with fiber an open cell of dimension -8\(\chi\)-2n.
For the basic results on convex \({\mathbb{R}}{\mathbb{P}}^ 2\)-structures on a closed S, see N. Kuiper [Conv. Int. Geom. Differ., Italia, 20-26 Sett. 1953, 200-213 (1954; Zbl 0057.143)], V. Kac and E. B. Vinberg [Math. Notes 1, 231-235 (1967); translation from Mat. Zametki 1, 347-354 (1967; Zbl 0163.169)] and J. P. Benzécri [Bull. Soc. Math. Fr. 88, 229-332 (1960; Zbl 0098.352)].
Reviewer: B.N.Apanasov


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57M50 General geometric structures on low-dimensional manifolds
30F60 Teichmüller theory for Riemann surfaces
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