## 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

### MSC:

 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

### Citations:

Zbl 0057.143; Zbl 0163.169; Zbl 0098.352
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