- Affine manifold
In
differential geometry , an affine manifold is amanifold 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 transformation s.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.
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