# Affine manifold

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 [math]\displaystyle{ {\mathbb R}^n }[/math], 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.

## Formal definition

An **affine manifold** [math]\displaystyle{ M\, }[/math] is a real manifold with charts [math]\displaystyle{ \psi_i\colon U_i\to{\mathbb R}^n }[/math] such that [math]\displaystyle{ \psi_i\circ\psi_j^{-1}\in \operatorname{Aff}({\mathbb R}^n) }[/math] for all [math]\displaystyle{ i, j\, , }[/math] where [math]\displaystyle{ \operatorname{Aff}({\mathbb R}^n) }[/math] denotes the Lie group of affine transformations. In fancier words it is a (G,X)-manifold where [math]\displaystyle{ X=\mathbb R^n }[/math] and [math]\displaystyle{ G }[/math] is the group of affine transformations.

An affine manifold is called **complete** if its universal covering is homeomorphic to [math]\displaystyle{ {\mathbb R}^n }[/math].

In the case of a compact affine manifold [math]\displaystyle{ M }[/math], let [math]\displaystyle{ G }[/math] be the fundamental group of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ \widetilde M }[/math] be its universal cover. One can show that each [math]\displaystyle{ n }[/math]-dimensional affine manifold comes with a developing map [math]\displaystyle{ D\colon {\widetilde M}\to{\mathbb R}^n }[/math], and a homomorphism [math]\displaystyle{ \varphi\colon G\to \operatorname{Aff}({\mathbb R}^n) }[/math], such that [math]\displaystyle{ D }[/math] is an immersion and equivariant with respect to [math]\displaystyle{ \varphi }[/math].

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.

## Important longstanding conjectures

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]}

## Notes

- ↑ 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**: 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. - ↑ "The Euler characteristic of an affine space form is zero".
*Bulletin of the American Mathematical Society***81**(5): 937–938. 1975. doi:10.1090/S0002-9904-1975-13896-1.

## References

- Nomizu, Katsumi; Sasaki, Takeshi (1994),
*Affine Differential Geometry*,*Cambridge University Press*, ISBN 978-0-521-44177-3, https://archive.org/details/affinedifferenti0000nomi - 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. https://archive.org/details/tensoranalysison00bish.

Original source: https://en.wikipedia.org/wiki/Affine manifold.
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