In the
mathematical
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 ...
field of
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 ...
, the exterior covariant derivative is an extension of the notion of
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 the setting of a differentiable
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 equip ...
or
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 ...
with a
connection.
Definition
Let ''G'' be a Lie group and be a
principal ''G''-bundle 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 ma ...
''M''. Suppose there is a
connection on ''P''; this yields a natural direct sum decomposition
of each tangent space into the
horizontal and
vertical
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
subspaces. Let
be the projection to the horizontal subspace.
If ''ϕ'' is a
''k''-form on ''P'' with values in a vector space ''V'', then its exterior covariant derivative ''Dϕ'' is a form defined by
:
where ''v''
''i'' are tangent vectors to ''P'' at ''u''.
Suppose that is a
representation of ''G'' on a vector space ''V''. If ''ϕ'' is
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
in the sense that
:
where
, then ''Dϕ'' is a
tensorial -form on ''P'' of the type ''ρ'': it is equivariant and horizontal (a form ''ψ'' is horizontal if .)
By
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
, the differential of ''ρ'' at the identity element may again be denoted by ''ρ'':
:
Let
be 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 ...
and
the representation of the connection in
That is,
is a
-valued form, vanishing on the horizontal subspace. If ''ϕ'' is a tensorial ''k''-form of type ''ρ'', then
:
where, following the notation in ', we wrote
:
Unlike the usual
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 ...
, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form ''ϕ'',
:
[Proof: Since ''ρ'' acts on the constant part of ''ω'', it commutes with ''d'' and thus
:.
Then, according to the example at ,
:
which is by E. Cartan's structure equation.]
where is the representation in
of the
curvature two-form Ω. The form F is sometimes referred to as 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 ...
, in analogy to the role it plays in
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 ...
. Note that ''D''
2 vanishes for a
flat connection In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie alge ...
(i.e. when ).
If , then one can write
:
where
is the matrix with 1 at the -th entry and zero on the other entries. The matrix
whose entries are 2-forms on ''P'' is called the curvature matrix.
For vector bundles
Given a smooth real vector bundle with a
connection and rank , the exterior covariant derivative is a real-linear map on the
vector-valued differential forms In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary diff ...
which are valued in :
:
The covariant derivative is such a map for . The exterior covariant derivatives extends this map to general . There are several equivalent ways to define this object:
* Suppose that a vector-valued differential 2-form is regarded as assigning to each a multilinear map which is completely anti-symmetric. Then the exterior covariant derivative assigns to each a multilinear map given by the formula
::
:where are arbitrary tangent vectors at which are extended to smooth locally-defined vector fields . The legitimacy of this definition depends on the fact that the above expression depends only on , and not on the choice of extension. This can be verified by the Leibniz rule for covariant differentiation and for the
Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth m ...
. The pattern established in the above formula in the case can be directly extended to define the exterior covariant derivative for arbitrary .
* The exterior covariant derivative may be characterized by the axiomatic property of defining for each a real-linear map which for is the covariant derivative and in general satisfies the Leibniz rule
::
:for any differential -form and any vector-valued form . This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential 1-form and any local frame of the vector bundle, the coordinates of are locally-defined differential 1-forms . The above inductive formula then says that
::
:In order for this to be a legitimate definition of , it must be verified that the choice of local frame is irrelevant. This can be checked by considering a second local frame obtained by an arbitrary change-of-basis matrix; the
inverse matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
provides the change-of-basis matrix for the 1-forms . When substituted into the above formula, the Leibniz rule as applied for the standard exterior derivative and for the covariant derivative cancel out the arbitrary choice.
* A vector-valued differential 2-form may be regarded as a certain collection of functions assigned to an arbitrary local frame of over a local coordinate chart of . The exterior covariant derivative is then defined as being given by the functions
::
:The fact that this defines a tensor field valued in is a direct consequence of the same fact for the covariant derivative. The further fact that it is a differential 3-form valued in asserts the full anti-symmetry in and is directly verified from the above formula and the contextual assumption that is a vector-valued differential 2-form, so that . The pattern in this definition of the exterior covariant derivative for can be directly extended to larger values of .
This definition may alternatively be expressed in terms of an arbitrary local frame of but without considering coordinates on . Then a vector-valued differential 2-form is expressed by differential 2-forms and the connection is expressed by the connection 1-forms, a skew-symmetric matrix of differential 1-forms . The exterior covariant derivative of , as a vector-valued differential 3-form, is expressed relative to the local frame by many differential 3-forms, defined by
::
In the case of the trivial real line bundle with its standard connection, vector-valued differential forms and differential forms can be naturally identified with one another, and each of the above definitions coincides with the standard
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 ...
.
Given a principal bundle, any
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
of the structure group defines an
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
, and any connection on the principal bundle induces a connection on the associated vector bundle. Differential forms valued in the vector bundle may be naturally identified with fully anti-symmetric
tensorial forms on the total space of the principal bundle. Under this identification, the notions of exterior covariant derivative for the principal bundle and for the vector bundle coincide with one another.
The
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives and , so that it is defined as a real-linear map . It is a fundamental but not immediately apparent fact that only depends on , and does so linearly. As such, the curvature may be regarded as an element of . Depending on how the exterior covariant derivative is formulated, various alternative but equivalent definitions of curvature (some without the language of exterior differentiation) can be obtained.
It is a well-known fact that the composition of the standard exterior derivative with itself is zero: . In the present context, this can be regarded as saying that the standard connection on the trivial line bundle has zero curvature.
Example
*
Bianchi's second identity In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie alge ...
, which says that the exterior covariant derivative of Ω is zero (that is, ) can be stated as:
.
Notes
References
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{{Tensors
Connection (mathematics)
Differential geometry
Fiber bundles