Affine Manifold
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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, an affine manifold is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
equipped with a flat, torsion-free
connection Connection may refer to: Mathematics *Connection (algebraic framework) *Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold * Connection (affine bundle) *Connection (composite bun ...
. Equivalently, it is a manifold that is (if connected)
covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of ...
by an open subset of ^n, with
monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
acting by
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
s. 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 A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent t ...
are
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
s (that is, have constant Jacobian matrix); 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 field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s form a
parallelisation Parallel computing is a type of computing, computation in which many calculations or Process (computing), processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. ...
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 In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
connection Connection may refer to: Mathematics *Connection (algebraic framework) *Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold * Connection (affine bundle) *Connection (composite bun ...
(also called
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
) and a
web Web most often refers to: * Spider web, a silken structure created by the animal * World Wide Web or the Web, an Internet-based hypertext system Web, WEB, or the Web may also refer to: Computing * WEB, a literate programming system created by ...
.


Formal definition

An affine manifold M\, is a real
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
with charts \psi_i\colon U_i\to^n such that \psi_i\circ\psi_j^\in \operatorname(^n) for all i, j\, , where \operatorname(^n) denotes the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
of
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
s. In fancier words it is a (G,X)-manifold where X=\mathbb R^n and G is the group of affine transformations. An affine manifold is called complete if its universal covering is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to ^n. In the case of a compact affine manifold M, let G be the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of M and \widetilde M be its
universal cover In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
. One can show that each n-dimensional affine manifold comes with a developing map D\colon \to^n, and a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
\varphi\colon G\to \operatorname(^n), such that D is an
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux * ''Immersion'' (album), the third album by Australian group Pendulum * ''Immersion'' (film), a 2021 ...
and equivariant with respect to \varphi. A
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of a compact complete flat affine manifold is called an affine
crystallographic group In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...
. 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 Ludwig Georg Elias Moses Bieberbach (; 4 December 1886 – 1 September 1982) was a German mathematician and leading representative of National Socialist German mathematics (" Deutsche Mathematik"). Biography Born in Goddelau, near Darmstadt, ...
, answering a question posed by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
. In his work on Hilbert's 18-th problem, Bieberbach proved that any Riemannian crystallographic group contains an abelian subgroup of finite index.


Complex affine manifolds

An ''affine complex manifold'' is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
that has an atlas whose
transition map In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies t ...
s belong to the group of complex
affine transformations In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally ...
, that is, have the form z \mapsto A \cdot z + c where n is the (complex) dimension of the manifold, c\in \mathbb C^n, and A is an invertible n \times n matrix with complex entries. In other words, it is a -manifold where X=\C^n and G is the group of complex affine transformations of \C^n.


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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it has parallel volume. Known in dimension 2. * Auslander conjecture (1964) stating that any affine crystallographic group contains a polycyclic subgroup of finite
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
. Known in dimensions up to 6, and when the
holonomy In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...
of the flat connection preserves a
Lorentz metric In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
. Since every virtually polycyclic crystallographic group preserves a
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
, Auslander conjecture implies the "only if" part of the Markus conjecture. * Chern conjecture (1955) The
Euler class In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
of an affine manifold vanishes.


Notes


References

* * * {{DEFAULTSORT:Affine Manifold Group theory Affine geometry Structures on manifolds Differential geometry Manifolds