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.
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.
is a real manifold with charts
denotes the group of affine transformations.
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
-dimensional affine manifold comes with a developing map
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.
An affine complex manifold is a complex manifold that has an atlas whose transition maps belong to the group of complex affine transformations, that is, have the form
is the (complex) dimension of the manifold,
matrix with complex entries.
is the group of complex affine transformations of
Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases.