Affine manifold

In differential geometry, an affine manifold is a manifold equipped with a flat, torsion-free connection.

Equivalently, it is a manifold that is (if connected) covered by an open subset of $\{Bbb\; R\}^n$, with monodromy acting by affine transformations.

Equivalently, it is a manifold equipped with an atlas, with all transition functions between charts affine; two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine.

Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases.

The most important of them are

*Markus conjecture (1961) stating that a compact affine manifold is complete if and only if it has constant volume. Known in dimension 3.

*Auslander conjecture (1964) stating that any group which acts properly and cocompactly on $\{Bbb\; R\}^n$ contains a polycyclic subgroup of finite index. Known in dimensions up to 8.

*Chern conjecture (1955) The Euler class of an affine manifold vanishes.

References

* Auslander L., "The structure of locally complete affine manifolds", Topology 3 (1964), 131-139.

* Fried D. and Goldman W., "Three dimensional affine chrystalographic groups", Adv. Math. 47 (1983), 1-49.

* Hirsch M. and Thurston W., "Foliated bundles, invariant measures, and flat manifolds," Ann. Math. (2) 101, (1975) 369-390.

* Konstant B., Sullivan D., "The Euler characteristic of an affine space form is zero," Bull. Amer. Math. Soc. 81 (1975), no. 5, 937-938.