Projective connection

In mathematics it is possible projective manifolds describe locally 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 1-dimensional subspaces of. The projective linear group acts as a group of invertible projective maps.

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

Specifically, the n- dimensional manifold has an open cover with homeomorphisms

Such that for all

The restriction of a mapping is off.

Analogously, one can define complex projective manifolds, here go the card images in the complex projective space and the map transitions are restrictions of maps in.

Convex projective manifolds

A projective manifold is convex projective if it is of the form of a convex subset and a discrete subgroup.

Examples

Hyperbolic manifolds

Hyperbolic manifolds are convex projective: the Beltrami -Klein model of hyperbolic space is a convex subset of the projective space, its isometry group is.

Flat manifolds are convex projective: the Euclidean space is a convex subset of the projective space, its isometry group is a subgroup of.

2-dimensional projective manifolds

Real projective structures on surfaces were classified by Choi and Goldman. The space of equivalence classes of real projective structures on a closed orientable surface of genus g is a countable union ( 16g -16) -dimensional open cells.

The moduli space of convex projective structures is a connected component in the Darstellungsvarietät the surface group.

3-dimensional projective manifolds

Theorem: Let a 3-manifold with one of the eight Thurston geometries. Then either is a non- orientable Seifert fibration ( and there is a 2- fold covering with a real projective structure) or the manifold has a unique, Thurston geometry underlying real projective structure.

This theorem follows from the representability of the Thurston geometries with the exception that in the case of product geometries and the group must be replaced by the group of orientation- preserving isometries, which is a subgroup of index 2.

In the case of non- orientable Seifert fibrations there is real projective structures that do not come from a projective representation of their Thurston geometry ( Guichard Vienna Hard). There are real projective structures on non-geometric 3-manifolds ( Benoist ), on the other hand, the connected sum no real projective structure ( Cooper Goldman ).