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The convex and concave decomposition of manifolds with real projective structures. (English) Zbl 0965.57016

This monograph collects, in a self-contained way, the more recent results in the study of flat real-projective structures on manifolds, ending with the description of a canonical decomposition. Such a decomposition involves manifolds with good convexity properties and manifolds with special affine structures. The absence of a metric, the presence of complicated global charts and the use of developing maps require a great deal of care in the investigation. Nevertheless, the obtained decomposition could be useful, for example, in the study of 3-manifolds with flat real projective structures. Namely every 3-manifold with homogeneous Riemannian structure has a natural (or a product) real projective structure, and the theory allows also the same author to give a classification of radiant affine 3-manifolds [S. Choi, Lect. Notes Ser., Seoul 46, 55-67 (1999; Zbl 0952.57005)].
The work begins, in Part I, with the basic tools on manifolds \(M\) modeled on (\(\mathbb{R}\text{P}^n, \text{GL}(n+1, \mathbb{R}))\), \(n\geq 2\), on convexity in the real projective sphere, on convexity in the Kuiper completions.
In Part II, the author gives Carrière’s notion of \(i\)-convexity, \(1\leq i\leq n-1\), a generalization of the usual convexity. He introduces some geometric objects as \(n\)-crescents, hemispherical \(n\)-crescents, bihedral \(n\)-crescents in the Kuiper completion of the universal covering of \(M\), and he studies their preservation under decomposition and splitting along two-faced submanifolds. Then, he proves the main theorem which states that a compact, real projective \(n\)-manifold, with empty or totally geodesic boundary, decomposes into real projective manifolds some of which are \((n-1)\)-convex and others are concave affine of special type. Finally, he states some consequences in the context of affine Lie groups.

MSC:

57N16 Geometric structures on manifolds of high or arbitrary dimension
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D99 Symplectic geometry, contact geometry
53A20 Projective differential geometry
57M50 General geometric structures on low-dimensional manifolds

Citations:

Zbl 0952.57005
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References:

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