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

Equivalently, it is a manifold that is (if connected) covered by an open subset of , with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem.

Equivalently, it is a manifold equipped with an atlas—called **the affine structure**—such that all transition functions between charts are affine transformations (that is, have constant Jacobian matrix);^{[1]} two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an **affine manifold** and the charts which are affinely related to those of the affine structure are called **affine charts**. In each affine coordinate domain the coordinate vector fields form a parallelisation of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection (also called affine connection) and a web.

An **affine manifold** is a real manifold with charts such that for all where denotes the Lie group of affine transformations. In fancier words it is a (G,X)-manifold where and is the group of affine transformations.

An affine manifold is called **complete** if its universal covering is homeomorphic to .

In the case of a compact affine manifold , let be the fundamental group of and be its universal cover. One can show that each -dimensional affine manifold comes with a developing map , and a homomorphism , such that is an immersion and equivariant with respect to .

A fundamental group of a compact complete flat affine manifold is called **an affine crystallographic group**. Classification of affine crystallographic groups is a difficult problem, far from being solved. The Riemannian crystallographic groups (also known as Bieberbach groups) were classified by Ludwig Bieberbach, answering a question posed by David Hilbert. In his work on Hilbert's 18-th problem, Bieberbach proved that any Riemannian crystallographic group contains an abelian subgroup of finite index.

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 (1962) stating that a compact affine manifold is complete if and only if it has parallel volume. Known in dimension 2.
- Auslander conjecture (1964)
^{[2]}^{[3]}stating that any affine crystallographic group contains a polycyclic subgroup of finite index. Known in dimensions up to 6,^{[4]}and when the holonomy of the flat connection preserves a Lorentz metric.^{[5]}Since every virtually polycyclic crystallographic group preserves a volume form, Auslander conjecture implies the "only if" part of the Markus conjecture.^{[6]} - Chern conjecture (1955) The Euler class of an affine manifold vanishes.
^{[7]}

**^**Bishop & Goldberg 1968, pp. 223–224.**^**Auslander, Louis (1964). "The structure of locally complete affine manifolds".*Topology*.**3**(Supplement 1): 131–139. doi:10.1016/0040-9383(64)90012-6.**^**Fried, Davis; Goldman, William M. (1983). "Three dimensional affine crystallographic groups".*Advances in Mathematics*.**47**(1): 1–49. doi:10.1016/0001-8708(83)90053-1.**^**Abels, Herbert; Margulis, Grigori A.; Soifer, Grigori A. (2002). "On the Zariski closure of the linear part of a properly discontinuous group of affine transformations".*Journal of Differential Geometry*.**60**(2): 315–344. doi:10.4310/jdg/1090351104.**^**Goldman, William M.; Kamishima, Yoshinobu (1984). "The fundamental group of a compact flat Lorentz space form is virtually polycyclic".*Journal of Differential Geometry*.**19**(1): 233–240. doi:10.4310/jdg/1214438430.**^**Abels, Herbert (2001). "Properly Discontinuous Groups of Affine Transformations: A Survey".*Geometriae Dedicata*.**87**: 309–333. doi:10.1023/A:1012019004745.**^**Kostant, Bertram; Sullivan, Dennis (1975). "The Euler characteristic of an affine space form is zero".*Bulletin of the American Mathematical Society*.**81**(5): 937–938. doi:10.1090/S0002-9904-1975-13896-1.

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*Affine Differential Geometry*, Cambridge University Press, ISBN 978-0-521-44177-3 - Sharpe, Richard W. (1997).
*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*. New York: Springer. ISBN 0-387-94732-9. - Bishop, Richard L.; Goldberg, Samuel I. (1968).
*Tensor Analysis on Manifolds*(First Dover 1980 ed.). The Macmillan Company. ISBN 0-486-64039-6.