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 linearconnection (also called affine connection) and a web.
Formal definition
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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.
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 .
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]
^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.