Projective manifold

This article deals with a class of locally homogeneous spaces. The term "projective manifolds" is occasionally used for projective varieties ; these are generally not projective manifolds for the purposes of this article.

In mathematics , projective manifolds can be described locally by projective coordinates. The projective manifolds include flat manifolds and hyperbolic manifolds and numerous other examples occurring in differential geometry and topology .

definition

The projective space is the space of the 1-dimensional subspace of the . The projective linear group acts as a group of invertible projective images . ${\displaystyle \mathbb {R} P^{n}}$ ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle PGL(n+1,\mathbb {R} )}$ ${\displaystyle \mathbb {R} P^{n}}$

A projective structure on a manifold is an atlas with map images in the projective space and projective images as map transitions.

More precisely: The n-dimensional manifold has an open coverage with homeomorphisms ${\displaystyle M}$ ${\displaystyle M=\cup _{i\in I}U_{i}}$

${\displaystyle \phi _{i}:U_{i}\rightarrow V_{i}\subset \mathbb {R} P^{n}}$,

so that for everyone ${\displaystyle i,j\in I}$

${\displaystyle \phi _{i}\phi _{j}^{-1}:\phi _{j}(U_{i}\cap U_{j})\rightarrow \phi _{i}(U_{i}\cap U_{j})}$

the limitation of a figure is off. ${\displaystyle PGL(n+1,\mathbb {R} )}$

Analogously, one can define complex projective manifolds, here the map images go into the complex projective space and the map transitions are restrictions of mappings in . ${\displaystyle \mathbb {C} P^{n}}$ ${\displaystyle PGL(n+1,\mathbb {C} )}$

Convex projective manifolds

A projective manifold is called convex projective if it is of the form for a convex subset and a discrete subgroup . ${\displaystyle M=\Gamma \backslash \Omega }$ ${\displaystyle \Omega \in \mathbb {R} P^{n}}$ ${\displaystyle \Gamma \subset PGL(n+1,\mathbb {R} )}$

Examples

Hyperbolic manifolds

Hyperbolic manifolds are convex projective: the Beltrami-Klein model of hyperbolic space is a convex subset of projective space, its isometric group is . ${\displaystyle PSO(n,1)\subset PGL(n+1,\mathbb {R} )}$

Flat manifolds

Flat manifolds are convex projective: Euclidean space is a convex subset of projective space, its isometric group is a subgroup of . ${\displaystyle PGL(n+1,\mathbb {R} )}$

2-dimensional projective manifolds

Really projective structures

Really projective structures on surfaces were classified by Choi and Goldman. The space of the equivalence classes of real projective structures on a closed, orientable surface of genus g is a countable union of (16g-16) -dimensional open cells. ${\displaystyle \Sigma _{g}}$

The modular space of the convex projective structures is a coherent component in the representation variety of the surface group , the Hitchin component . ${\displaystyle Hom(\pi _{1}\Sigma _{g},PGL(3,\mathbb {R} ))}$ ${\displaystyle \pi _{1}\Sigma _{g}}$

Complex projective structures

All complex projective structures on surfaces can be constructed from hyperbolic structures by "grafting" along measured laminations .

3-dimensional projective manifolds

Theorem : Let be a 3-manifold with one of the 8 Thurston geometries . Then there is either a non-orientable Seifert fiber (and there is a 2-fold overlay with a real projective structure) or the manifold has an unambiguous, real projective structure on which the Thurston geometry is based. ${\displaystyle M}$ ${\displaystyle M}$

This sentence follows from the representability of the Thurston geometries in with the exception that in the case of the product geometries and the group must be replaced by the group of the orientation-maintaining isometrics, which is a subgroup of the index 2. ${\displaystyle (X,G)}$ ${\displaystyle (\mathbb {R} P^{3},PGL(4,\mathbb {R} ))}$ ${\displaystyle S^{2}\times \mathbb {R} }$ ${\displaystyle H^{2}\times \mathbb {R} }$ ${\displaystyle G=Isom(X)}$ ${\displaystyle Isom^{+}(X)}$

In the case of non-orientable Seifert fibers, there are real projective structures that do not come from a projective representation of their Thurston geometry (Guichard-Wienhard). There are real projective structures also on non-geometric 3-manifolds (Benoist), on the other hand the connected sum can not have a real projective structure (Cooper-Goldman). ${\displaystyle \mathbb {R} P^{3}\sharp \mathbb {R} P^{3}}$

literature

• Choi, Suhyoung; Goldman, William M .: The classification of real projective structures on compact surfaces. Bull. Amer. Math. Soc. (NS) 34 (1997), no. 2, 161-171.
• Cooper, Daryl; Goldman, William M .: A 3-manifold with no real projective structure. http://arxiv.org/abs/1207.2007
• Goldman, William M. What is… a projective structure? Notices Amer. Math. Soc. 54 (2007), no. 1, 30-33. pdf

Web links

• Sam Ballas: Flexibility and rigidity of 3-dimensional convex projective structures

Individual evidence

1. ↑ Choi, Suhyoung; Goldman, William M. Convex real projective structures on closed surfaces are closed. Proc. Amer. Math. Soc. 118 (1993) no. 2, 657-661.
2. ↑ Kamishima, Yoshinobu; Tan, Ser P. Deformation spaces on geometric structures. Aspects of low-dimensional manifolds, 263-299, Adv. Stud. Pure Math., 20, Kinokuniya, Tokyo, 1992.