In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, if ''X'' is a manifold with an action of a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
''G'' by analytical diffeomorphisms, the notion of a (''G'', ''X'')-structure on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is a way to formalise it being locally isomorphic to ''X'' with its ''G''-invariant structure; spaces with a (''G'', ''X'')-structure are always
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 ...
s and are called (''G'', ''X'')-manifolds. This notion is often used with ''G'' being a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
and ''X'' a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
for ''G''. Foundational examples are
hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, r ...
s and
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 ^n, with monodromy acting by affine tr ...
s.
Definition and examples
Formal definition
Let
be a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
differential 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 ma ...
and
be a subgroup of the group of
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
s of
which act analytically in the following sense:
:if
and there is a nonempty open subset
such that
are equal when restricted to
then
(this definition is inspired by the analytic continuation property of analytic diffeomorphisms on an
analytic manifold
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic ge ...
).
A
-structure on a topological space
is a
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 ...
structure on
whose
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
' charts has values in
and
transition maps belong to
. This means that there exists:
*a covering of
by open sets
(i.e.
);
*open
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
s
called charts;
such that every transition map
is the restriction of a diffeomorphism in
.
Two such structures
are equivalent when they are contained in a maximal one, equivalently when their union is also a
structure (i.e. the maps
and
are restrictions of diffeomorphisms in
).
Riemannian examples
If
is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
and
a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
with a
faithful action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of
by
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
then the action is analytic. Usually one takes
to be the full isometry group of
. Then the category of
manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to
(i.e. every point has a neighbourhood isometric to an open subset of
).
Often the examples of
are
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
under
, for example one can take
with a left-invariant metric. A particularly simple example is
and
the group of
euclidean isometries. Then a
manifold is simply a
flat manifold In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles ...
.
A particularly interesting example is when
is a Riemannian
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
, for example
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
. The simplest such example is the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
, whose isometry group is isomorphic to
.
Pseudo-Riemannian examples
When
is
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
and
the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
the notion of a
-structure is the same as that of a flat
Lorentzian manifold
In differential geometry, 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 ...
.
Other examples
When
is the affine space and
the group of affine transformations then one gets the notion of an
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 ^n, with monodromy acting by affine tr ...
.
When
is the n-dimensional real
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
and
one gets the notion of a projective structure.
Developing map and completeness
Developing map
Let
be a
-manifold which is connected (as a topological space). The developing map is a map from the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
to
which is only well-defined up to composition by an element of
.
A developing map is defined as follows: fix
and let
be any other point,
a path from
to
, and
(where
is a small enough neighbourhood of
) a map obtained by composing a chart of
with the projection
. We may use analytic continuation along
to extend
so that its domain includes
. Since
is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
the value of
thus obtained does not depend on the original choice of
, and we call the (well-defined) map
a ''developing map'' for the
-structure. It depends on the choice of base point and chart, but only up to composition by an element of
.
Monodromy
Given a developing map
, the ''monodromy'' or ''holonomy'' of a
-structure is the unique morphism
which satisfies
:
.
It depends on the choice of a developing map but only up to an
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
of
.
Complete (''G'',''X'')-structures
A
structure is said to be ''complete'' if it has a developing map which is also a
covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete sp ...
(this does not depend on the choice of developing map since they differ by a diffeomorphism). For example, if
is simply connected the structure is complete if and only if the developing map is a diffeomorphism.
Examples
Riemannian (''G'',''X'')-structures
If
is a Riemannian manifold and
its full group of isometry, then a
-structure is complete if and only if the underlying Riemannian manifold is
geodesically complete In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map a ...
(equivalently metrically complete). In particular, in this case if the underlying space of a
-manifold is compact then the latter is automatically complete.
In the case where
is the hyperbolic plane the developing map is the same map as given by the
Uniformisation Theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalizatio ...
.
Other cases
In general compactness of the space does not imply completeness of a
-structure. For example, an affine structure on the torus is complete if and only if the monodromy map has its image inside the
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. But there are many affine tori which do not satisfy this condition, for example any quadrilateral with its opposite sides glued by an affine map yields an affine structure on the torus, which is complete if and only if the quadrilateral is a parallelogram.
Interesting examples of complete, noncompact affine manifolds are given by the Margulis spacetimes.
(''G'',''X'')-structures as connections
In the work of
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differential ...
-structures on a manifold
are viewed as flat
Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it d ...
s on
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s with fiber
over
, whose
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 ...
maps lie in
.
Notes
References
*
{{DEFAULTSORT:G X manifold
Differential geometry
Structures on manifolds