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 ...
, a metric connection is a
connection in 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 ...
''E'' equipped with a
bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.
Definition
If ''M'' is a topological manifold ...
; that is, a metric for which the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of any two vectors will remain the same when those vectors are
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 ...
ed along any curve.
[
.(''Third edition: see chapter 3; Sixth edition: see chapter 4.'')
] This is equivalent to:
* A connection for which the
covariant derivatives of the metric on ''E'' vanish.
* A
principal connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-conne ...
on the bundle of
orthonormal frame
In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If ''M'' is a manifold equipped with a metric ''g'', then an orthonormal frame at a point ...
s of ''E''.
A special case of a metric connection is a
Riemannian connection
In mathematics, a metric connection is a connection (vector bundle), connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are p ...
; there is a unique such which is
torsion free, 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 ...
. In this case, the bundle ''E'' is 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 ...
''TM'' of a manifold, and the metric on ''E'' is induced by a Riemannian metric on ''M''.
Another special case of a metric connection is a
Yang–Mills connection, which satisfies the
Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the E ...
of motion. Most of the machinery of defining a connection and its curvature can go through without requiring any compatibility with the bundle metric. However, once one does require compatibility, this metric connection defines an inner product,
Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
,
Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
, and
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, which are required to formulate the Yang–Mills equations.
Definition
Let
be any
local sections of the vector bundle ''E'', and let ''X'' be a vector field on the base space ''M'' of the bundle. Let
define a
bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.
Definition
If ''M'' is a topological manifold ...
, that is, a metric on the vector fibers of ''E''. Then, a
connection ''D'' on ''E'' is a metric connection if:
:
Here ''d'' is the ordinary
differential of a scalar function. The covariant derivative can be extended so that it acts as a map on ''E''-valued
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
on the base space:
:
One defines
for a function
, and
:
where
is a local smooth section for the vector bundle and
is a (scalar-valued) ''p''-form. The above definitions also apply to
local smooth frames as well as local sections.
Metric versus dual pairing
The bundle metric
imposed on ''E'' should not be confused with the natural pairing
of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s
so that
:
pairs vectors with dual vectors (functionals) above each point of ''M''. That is, if
is any local coordinate frame on ''E'', then one naturally obtains a dual coordinate frame
on ''E''* satisfying
.
By contrast, the bundle metric
is a function on
:
giving an inner product on each vector space fiber of ''E''. The bundle metric allows one to define an ''orthonormal'' coordinate frame by the equation
Given a vector bundle, it is always possible to define a bundle metric on it.
Following standard practice,
[ one can define a ]connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
, 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 ...
and the Riemann curvature
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rig ...
without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity. However, to define the Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
, the Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, the first Bianchi identity, and the Yang–Mills functional, one needs the bundle metric.
Connection form
Given a local bundle chart, the covariant derivative can be written in the form
:
where ''A'' is the connection one-form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Car ...
.
A bit of notational machinery is in order. Let denote the space of differentiable sections on ''E'', let denote the space of ''p''-forms on ''M'', and let be the endomorphisms on ''E''. The covariant derivative, as defined here, is a map
:
One may express the connection form in terms of the connection coefficient
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 ...
s as
:
The point of the notation is to distinguish the indices ''j'', ''k'', which run over the ''n'' dimensions of the fiber, from the index ''i'', which runs over the ''m''-dimensional base space. For the case of a Riemann connection below, the vector space ''E'' is taken to be the tangent bundle ''TM'', and .
The notation of ''A'' for the connection form comes from physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, in historical reference to the vector potential field of electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
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) ...
. In mathematics, the notation is often used in place of ''A'', as in the article on the connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
; unfortunately, the use of for the connection form collides with the use of to denote a generic alternating form
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
on the vector bundle.
Skew symmetry
The connection is skew-symmetric in the vector-space (fiber) indices; that is, for a given vector field , the matrix is skew-symmetric; equivalently, it is an element of the Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
.
This can be seen as follows. Let the fiber be ''n''-dimensional, so that the bundle ''E'' can be given an orthonormal local frame with . One then has, by definition, that , so that:
:
In addition, for each point of the bundle chart, the local frame is orthonormal:
:
It follows that, for every vector , that
:
That is, is skew-symmetric.
This is arrived at by explicitly using the bundle metric; without making use of this, and using only the pairing , one can only relate the connection form ''A'' on ''E'' to its dual ''A'' on ''E'', as This follows from the ''definition'' of the dual connection as
Curvature
There are several notations in use for the curvature of a connection, including a modern one using ''F'' to denote the field strength tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, a classical one using ''R'' as the curvature tensor, and the classical notation for the Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, most of which can be extended naturally to the case of vector bundles. ''None'' of these definitions require either a metric tensor, or a bundle metric, and can be defined quite concretely without reference to these. The definitions do, however, require a clear idea of the endomorphisms of ''E'', as described above.
Compact style
The most compact definition of the curvature ''F'' is to define it as the 2-form taking values in , given by the amount by which the connection fails to be exact; that is, as
:
which is an element of
:
or equivalently,
:
To relate this to other common definitions and notations, let be a section on ''E''. Inserting into the above and expanding, one finds
:
or equivalently, dropping the section
:
as a terse definition.
Component style
In terms of components, let where is the standard one-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
coordinate bases on the cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
''T''*''M''. Inserting into the above, and expanding, one obtains (using the summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
):
:
Keep in mind that for an ''n''-dimensional vector space, each is an ''n''×''n'' matrix, the indices of which have been suppressed, whereas the indices ''i'' and ''j'' run over 1,...,''m'', with ''m'' being the dimension of the underlying manifold. Both of these indices can be made simultaneously manifest, as shown in the next section.
The notation presented here is that which is commonly used in physics; for example, it can be immediately recognizable as the gluon field strength tensor
In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.
The strong interaction is one of the fundamental interactions of nature, and the quantum fie ...
. For the abelian case, ''n''=1, and the vector bundle is one-dimensional; the commutator vanishes, and the above can then be recognized as the electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
in more or less standard physics notation.
Relativity style
All of the indices can be made explicit by providing a 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 ...
, on . A given section then may be written as
:
In this local frame, the connection form becomes
:
with being the Christoffel symbol
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 distan ...
; again, the index ''i'' runs over (the dimension of the underlying manifold ''M'') while ''j'' and ''k'' run over , the dimension of the fiber. Inserting and turning the crank, one obtains
:
where now identifiable as the Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
. This is written in the style commonly employed in many textbooks on 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 ...
from the middle-20th century (with several notable exceptions, such as MTW, that pushed early on for an index-free notation). Again, the indices ''i'' and ''j'' run over the dimensions of the manifold ''M'', while ''r'' and ''k'' run over the dimension of the fibers.
Tangent-bundle style
The above can be back-ported to the vector-field style, by writing as the standard basis elements for 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 ...
''TM''. One then defines the curvature tensor as
:
so that the spatial directions are re-absorbed, resulting in the notation
:
Alternately, the spatial directions can be made manifest, while hiding the indices, by writing the expressions in terms of vector fields ''X'' and ''Y'' on ''TM''. In the standard basis, ''X'' is
:
and likewise for ''Y''. After a bit of plug and chug
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, ...
, one obtains
:
where
:
is the Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of the vector field ''Y'' with respect to ''X''.
To recap, the curvature tensor maps fibers to fibers:
:
so that
:
To be very clear, are alternative notations for the same thing. Observe that none of the above manipulations ever actually required the bundle metric to go through. One can also demonstrate the second Bianchi identity
:
without having to make any use of the bundle metric.
Yang–Mills connection
The above development of the curvature tensor did not make any appeals to the bundle metric. That is, they did not need to assume that ''D'' or ''A'' were metric connections: simply having a connection on a vector bundle is sufficient to obtain the above forms. All of the different notational variants follow directly only from consideration of the endomorphisms of the fibers of the bundle.
The bundle metric is required to define the Hodge star
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
and the Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
; that is needed, in turn, to define the Laplacian, and to demonstrate that
:
Any connection that satisfies this identity is referred to as a Yang–Mills connection. It can be shown that this connection is a critical point of the Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s applied to the Yang–Mills action
:
where is the volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
, the Hodge dual
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
of the constant 1. Note that three different inner products are required to construct this action: the metric connection on ''E'', an inner product on End(''E''), equivalent to the quadratic Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the Center (ring theory), center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared ang ...
(the trace of a pair of matricies), and the Hodge dual.
Riemannian connection
An important special case of a metric connection is a Riemannian connection. This is a connection 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 ...
(''M'', ''g'') such that for all vector fields ''X'' on ''M''. Equivalently, is Riemannian if the 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 ...
it defines preserves the metric ''g''.
A given connection is Riemannian if and only if
:
for all vector fields ''X'', ''Y'' and ''Z'' on ''M'', where denotes the derivative of the function along this vector field .
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 the torsion-free Riemannian connection on a manifold. It is unique by the fundamental theorem of Riemannian geometry
In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatibl ...
. For every Riemannian connection, one may write a (unique) corresponding Levi-Civita connection. The difference between the two is given by the contorsion tensor The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subje ...
.
In component notation, the covariant derivative is compatible with the metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
if
:
Although other covariant derivatives may be defined, usually one only considers the metric-compatible one. This is because given two covariant derivatives, and , there exists a tensor for transforming from one to the other:
:
If the space is also torsion-free, then the tensor is symmetric in its first two indices.
A word about notation
It is conventional to change notation and use the nabla symbol ∇ in place of ''D'' in this setting; in other respects, these two are the same thing. That is, ∇ = ''D'' from the previous sections above.
Likewise, the inner product on ''E'' is replaced by the metric tensor ''g'' on ''TM''. This is consistent with historic usage, but also avoids confusion: for the general case of a vector bundle ''E'', the underlying manifold ''M'' is ''not'' assumed to be endowed with a metric. The special case of manifolds with both a metric ''g'' on ''TM'' in addition to a bundle metric on ''E'' leads to Kaluza–Klein theory.
See also
* Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the vertical bundle VE and horizontal bundle HE are subbundles of ...
References
*
*
*
{{tensors
Connection (mathematics)
Riemannian geometry