
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 ...
, the holonomy of a
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 ...
on a
smooth 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 may ...
is the extent to which
parallel transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
of the connection. For flat connections, the associated holonomy is a type of
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 ...
and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.
Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
. Important examples include: holonomy of the
Levi-Civita connection
In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
(called Riemannian holonomy), holonomy of
connections
Connections may refer to:
* Connection (disambiguation), plural form
Television
* '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series
* ''Connections'' (British TV series), a 1978 documentary tele ...
in
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s, holonomy of
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
s, and holonomy of
connections
Connections may refer to:
* Connection (disambiguation), plural form
Television
* '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series
* ''Connections'' (British TV series), a 1978 documentary tele ...
in
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
s. In each of these cases, the holonomy of the connection can be identified with a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
, the holonomy group. The holonomy of a connection is closely related to the curvature of the connection, via the ''
Ambrose–Singer theorem
In differential geometry, the holonomy of a connection (mathematics), 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 genera ...
''.
The study of Riemannian holonomy has led to a number of important developments. Holonomy was introduced by in order to study and classify
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...
s. It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. In 1952
Georges de Rham
Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
Biography
Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in ...
proved the ''de Rham decomposition theorem'', a principle for splitting a Riemannian manifold into a
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of Riemannian manifolds by splitting the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
into irreducible spaces under the
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
of the local holonomy groups. Later, in 1953,
Marcel Berger
Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France.
Biography
After studying from 1948 to 19 ...
classified the possible irreducible holonomies. The decomposition and classification of Riemannian holonomy has applications to physics and to
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
.
Definitions
Holonomy of a connection in a vector bundle
Let ''E'' be a rank-''k''
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
over a
smooth 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 may ...
''M'', and let ∇ be a
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 ...
on ''E''. Given a
piecewise
In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...
smooth
loop ''γ'' :
,1→ ''M'' based at ''x'' in ''M'', the connection defines a
parallel transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
map ''P''
''γ'' : ''E
x'' → ''E
x'' on the fiber of ''E'' at ''x''. This map is both linear and invertible, and so defines an element of the
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
GL(''E
x''). The holonomy group of ∇ based at ''x'' is defined as
:
The restricted holonomy group based at ''x'' is the subgroup
coming from
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...
loops ''γ''.
If ''M'' is
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 ...
, then the holonomy group depends on the
basepoint ''x'' only
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
*Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change o ...
in GL(''k'', R). Explicitly, if ''γ'' is a path from ''x'' to ''y'' in ''M'', then
:
Choosing different identifications of ''E
x'' with R
''k'' also gives conjugate subgroups. Sometimes, particularly in general or informal discussions (such as below), one may drop reference to the basepoint, with the understanding that the definition is good up to conjugation.
Some important properties of the holonomy group include:
*
is a connected
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
of GL(''k'', R).
*
is the
identity component
In mathematics, specifically group theory, the identity component of a group (mathematics) , group ''G'' (also known as its unity component) refers to several closely related notions of the largest connected space , connected subgroup of ''G'' co ...
of
* There is a natural,
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
where
is 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'', which sends the homotopy class