In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, a connection on a
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 ...
is a device that defines a notion of
parallel transport
In 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 (vector bundle), c ...
on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a
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 every po ...
, for which the notion of parallel transport must be
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
. A linear connection is equivalently specified by a ''
covariant derivative'', an operator that differentiates
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the bundle along
tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, 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 (i.e. affine connection) that preserves th ...
on the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a
pseudo-Riemannian 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 ...
, which gives a standard way to differentiate vector fields.
Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
Linear connections are also called Koszul connections after
Jean-Louis Koszul
Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki.
Biography
Koszul was educated at the in ...
, who gave an algebraic framework for describing them .
This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, vector bundle computations are usually written using indexed tensors; in
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on
metric connections (the comments made there apply to all vector bundles).
Motivation
Let be 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 ma ...
, such as
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. A vector-valued function
can be viewed as a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of the trivial
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 every po ...
One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on .
The model case is to differentiate a function
on Euclidean space
. In this setting the derivative
at a point
in the direction
may be defined by the standard formula
:
For every
, this defines a new vector
When passing to a section
of a vector bundle
over a manifold
, one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term
makes no sense on
. Instead one takes a path
such that
and computes
:
However this still does not make sense, because
and
are elements of the distinct vector spaces
and
This means that subtraction of these two terms is not naturally defined.
The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.
# (''
Parallel transport
In 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 (vector bundle), c ...
'') A connection can be viewed as assigning to every differentiable path
a
linear isomorphism
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
for all
Using this isomorphism one can transport
to the fibre
and then take the difference; explicitly,
In order for this to depend only on
and not on the path
extending
it is necessary to place restrictions (in the definition) on the dependence of
on
This is not straightforward to formulate, and so this notion of "parallel transport" is usually derived as a by-product of other ways of defining connections. In fact, the following notion of "Ehresmann connection" is nothing but an infinitesimal formulation of parallel transport.
# (''
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 ...
'') The section
may be viewed as a smooth map from the smooth manifold
to the smooth manifold
As such, one may consider the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
which is an element of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
In Ehresmann's formulation of a connection, one chooses a way of assigning, to each
and every
a direct sum decomposition of
into two linear subspaces, one of which is the natural embedding of
With this additional data, one defines
by projecting
to be valued in
In order to respect the linear structure of a vector bundle, one imposes additional restrictions on how the direct sum decomposition of
moves as is varied over a fiber.
# (''
Covariant derivative'') The standard derivative
in Euclidean contexts satisfies certain dependencies on
and
the most fundamental being linearity. A covariant derivative is defined to be any operation
which mimics these properties, together with a form of the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
.
Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a corresponding ''choice'' of how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certain
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. In the case of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
, any
pseudo-Riemannian metric
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 ...
(and in particular any
Riemannian metric
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 ''T ...
) determines a canonical connection, called 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 (i.e. affine connection) that preserves th ...
.
Formal definition
Let
be a smooth real
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 every po ...
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 ma ...
. Denote the space of smooth
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of
by
. A covariant derivative on
is either of the following equivalent structures:
# an
-
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
such that the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
holds for all
smooth functions
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
on
and all smooth sections
of
# an assignment, to any smooth section and every
, of a
-linear map
which depends smoothly on and such that
for any two smooth sections
and any real numbers
and such that for every smooth function
,
is related to
by
for any
and
Beyond using the canonical identification between the vector space
and the vector space of linear maps
these two definitions are identical and differ only in the language used.
It is typical to denote
by
with
being implicit in
With this notation, the product rule in the second version of the definition given above is written
:
''Remark.'' In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "ℝ" everywhere they appear to "complex" and "
" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.
Induced connections
Given a vector bundle
, there are many associated bundles to
which may be constructed, for example the dual vector bundle
, tensor powers
, symmetric and antisymmetric tensor powers
, and the direct sums
. A connection on
induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory of
principal bundle connections, but here we present some of the basic induced connections.
Dual connection
Given
a connection on
, the induced dual connection
on
is defined implicitly by
:
Here
is a smooth vector field,
is a section of
, and
a section of the dual bundle, and
the natural pairing between a vector space and its dual (occurring on each fibre between
and
), i.e.,
. Notice that this definition is essentially enforcing that
be the connection on
so that a natural
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
is satisfied for pairing
.
Tensor product connection
Given
connections on two vector bundles
, define the tensor product connection by the formula
:
Here we have
. Notice again this is the natural way of combining
to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor product
, one also obtains the tensor power connection on
for any
and vector bundle
.
Direct sum connection
The direct sum connection is defined by
:
where
.
Symmetric and exterior power connections
Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power,
, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
as direct summands, and the connection
respects this natural splitting, one can simply restrict
to these summands. Explicitly, define the symmetric product connection by
:
and the exterior product connection by
:
for all
. Repeated applications of these products gives induced symmetric power and exterior power connections on
and
respectively.
Endomorphism connection
Finally, one may define the induced connection
on the vector bundle of endomorphisms
, the endomorphism connection. This is simply the tensor product connection of the dual connection
on
and
on
. If
and
, so that the composition
also, then the following product rule holds for the endomorphism connection:
:
By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying
:
for any
, thus avoiding the need to first define the dual connection and tensor product connection.
Any associated bundle
Given a vector bundle
of rank
, and any representation
into a linear group
, there is an induced connection on the associated vector bundle
. This theory is most succinctly captured by passing to the principal bundle connection on the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
of
and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.
Exterior covariant derivative and vector-valued forms
Let
be a vector bundle. An
-valued differential form of degree
is a section of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
bundle:
:
The space of such forms is denoted by
:
where the last tensor product denotes the tensor product of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of smooth functions on
.
An
-valued 0-form is just a section of the bundle
. That is,
:
In this notation a connection on
is a linear map
:
A connection may then be viewed as a generalization of the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
to vector bundle valued forms. In fact, given a connection
on
there is a unique way to extend
to an
exterior covariant derivative
In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.
Definition
Let ''G' ...
:
This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the form
and extended linearly:
:
where
so that
,
is a section, and
denotes the
-form with values in
defined by wedging
with the one-form part of
. Notice that for
-valued 0-forms, this recovers the normal Leibniz rule for the connection
.
Unlike the ordinary exterior derivative, one generally has
. In fact,
is directly related to the curvature of the connection
(see
below).
Affine properties of the set of connections
Every vector bundle over a manifold admits a connection, which can be proved using
partitions of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
. However, connections are not unique. If
and
are two connections on
then their difference is a
-linear operator. That is,
:
for all smooth functions
on
and all smooth sections
of
. It follows that the difference
can be uniquely identified with a one-form on
with values in the endomorphism bundle
:
:
Conversely, if
is a connection on
and
is a one-form on
with values in
, then
is a connection on
.
In other words, the space of connections on
is an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
for
. This affine space is commonly denoted
.
Relation to principal and Ehresmann connections
Let
be a vector bundle of rank
and let
be the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
of
. Then a
(principal) connection on
induces a connection on
. First note that sections of
are in one-to-one correspondence with
right-equivariant maps
. (This can be seen by considering the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
of
over
, which is isomorphic to the
trivial 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 ...
.) Given a section
of
let the corresponding equivariant map be
. The covariant derivative on
is then given by
:
where
is the
horizontal lift of
from
to
. (Recall that the horizontal lift is determined by the connection on
.)
Conversely, a connection on
determines a connection on
, and these two constructions are mutually inverse.
A connection on
is also determined equivalently by a
linear Ehresmann connection on
. This provides one method to construct the associated principal connection.
The induced connections discussed in
#Induced connections can be constructed as connections on other associated bundles to the frame bundle of
, using representations other than the standard representation used above. For example if
denotes the standard representation of
on
, then the associated bundle to the representation
of
on
is the direct sum bundle
, and the induced connection is precisely that which was described above.
Local expression
Let
be a vector bundle of rank
, and let
be an open subset of
over which
trivialises. Therefore over the set
,
admits a local
smooth frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay t ...
of sections
:
Since the frame
defines a basis of the fibre
for any
, one can expand any local section
in the frame as
:
for a collection of smooth functions
.
Given a connection
on
, it is possible to express
over
in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis section
, the quantity
may be expanded in the local frame
as
:
where
are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by
:
called the ''local connection form of
over
''. The action of
on any section
can be computed in terms of
using the product rule as
:
If the local section
is also written in matrix notation as a column vector using the local frame
as a basis,
:
then using regular matrix multiplication one can write
:
where
is shorthand for applying the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
to each component of
as a column vector. In this notation, one often writes locally that
. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.
As explained in
#Affine properties of the set of connections, any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-form
is precisely the endomorphism-valued one-form such that the connection
on
differs from the trivial connection
on
, which exists because
is a trivialising set for
.
Relationship to Christoffel symbols
In
pseudo-Riemannian geometry
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 t ...
, 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 (i.e. affine connection) that preserves th ...
is often written in terms of the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
instead of the connection one-form
. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition to
being a trivialising open subset for the vector bundle
, that
is also a
local chart for the manifold
, admitting local coordinates
.
In such a local chart, there is a distinguished local frame for the differential one-forms given by
, and the local connection one-forms
can be expanded in this basis as
:
for a collection of local smooth functions
, called the ''Christoffel symbols'' of
over
. In the case where
and
is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.
The expression for how
acts in local coordinates can be further expanded in terms of the local chart
and the Christoffel symbols, to be given by
:
Contracting this expression with the local coordinate tangent vector
leads to
:
This defines a collection of
locally defined operators
:
with the property that
:
Change of local trivialisation
Suppose
is another choice of local frame over the same trivialising set
, so that there is a matrix
of smooth functions relating
and
, defined by
:
Tracing through the construction of the local connection form
for the frame
, one finds that the connection one-form
for
is given by
:
where
denotes the inverse matrix to
. In matrix notation this may be written
:
where
is the matrix of one-forms given by taking the exterior derivative of the matrix
component-by-component.
In the case where
is the tangent bundle and
is the Jacobian of a coordinate transformation of
, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.
Parallel transport and holonomy
A connection
on a vector bundle
defines a notion of
parallel transport
In 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 (vector bundle), c ...
on
along a curve in
. Let
be a smooth
path
A path is a route for physical travel – see Trail.
Path or PATH may also refer to:
Physical paths of different types
* Bicycle path
* Bridle path, used by people on horseback
* Course (navigation), the intended path of a vehicle
* Desire p ...
in
. A section
of
along
is said to be parallel if
:
for all