Exterior covariant derivative

Template:See also 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 be a Lie group and Template:Nowrap be a principal G-bundle on a smooth manifold M. Suppose there is a connection on P; this yields a natural direct sum decomposition <math>T_u P = H_u \oplus V_u</math> of each tangent space into the horizontal and vertical subspaces. Let <math>h: T_u P \to H_u</math> 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

<math>D\phi(v_0, v_1,\dots, v_k)= d \phi(h v_0 ,h v_1,\dots, h v_k)</math>

where v_i are tangent vectors to P at u.

Suppose that Template:Nowrap is a representation of G on a vector space V. If ϕ is equivariant in the sense that

<math>R_g^* \phi = \rho(g)^{-1}\phi</math>

where <math>R_g(u) = ug</math>, then Dϕ is a [[tensorial form|tensorial Template:Nowrap-form]] on P of the type ρ: it is equivariant and horizontal (a form ψ is horizontal if Template:Nowrap.)

By abuse of notation, the differential of ρ at the identity element may again be denoted by ρ:

<math>\rho: \mathfrak{g} \to \mathfrak{gl}(V).</math>

Let <math>\omega</math> be the connection one-form and <math>\rho(\omega)</math> the representation of the connection in <math>\mathfrak{gl}(V).</math> That is, <math>\rho(\omega)</math> is a <math>\mathfrak{gl}(V)</math>-valued form, vanishing on the horizontal subspace. If ϕ is a tensorial k-form of type ρ, then

<math>D \phi = d \phi + \rho(\omega) \cdot \phi,</math><ref>If Template:Nowrap, then, writing <math>X^{\#}</math> for the fundamental vector field (i.e., vertical vector field) generated by X in <math>\mathfrak{g}</math> on P, we have:

<math>d \phi(X^{\#}_u) = \left . {d \over dt}\right\vert_0 \phi(u \operatorname{exp}(tX)) = -\rho(X)\phi(u) = -\rho(\omega(X^{\#}_u))\phi(u)</math>,

since Template:Nowrap. On the other hand, Template:Nowrap. If X is a horizontal tangent vector, then <math>D \phi(X) = d\phi(X)</math> and <math>\omega(X) = 0</math>. For the general case, let X_i's be tangent vectors to P at some point such that some of X_i's are horizontal and the rest vertical. If X_i is vertical, we think of it as a Lie algebra element and then identify it with the fundamental vector field generated by it. If X_i is horizontal, we replace it with the horizontal lift of the vector field extending the pushforward πX_i. This way, we have extended X_i's to vector fields. Note the extension is such that we have: [X_i, X_j] = 0 if X_i is horizontal and X_j is vertical. Finally, by the invariant formula for exterior derivative, we have:

<math>D \phi(X_0, \dots, X_k) - d \phi(X_0, \dots, X_k) = {1 \over k+1} \sum_0^k (-1)^i \rho(\omega(X_i)) \phi(X_0, \dots, \widehat{X_i}, \dots, X_k)</math>,

which is <math>(\rho(\omega) \cdot \phi)(X_0, \cdots, X_k)</math>.</ref>

where, following the notation in Template:Section link, we wrote

<math>

 (\rho(\omega) \cdot \phi)(v_1, \dots, v_{k+1}) =
 {1 \over (1+k)!} \sum_{\sigma}  \operatorname{sgn}(\sigma)\rho(\omega(v_{\sigma(1)})) \phi(v_{\sigma(2)}, \dots, v_{\sigma(k+1)}).

</math>

Unlike the usual exterior derivative, which squares to 0, the exterior covariant derivative does not. In general, one has, for a tensorial zero-form ϕ,

<math>D^2\phi=F \cdot \phi.</math><ref>Proof: Since ρ acts on the constant part of ω, it commutes with d and thus

<math>d(\rho(\omega) \cdot \phi) = d(\rho(\omega)) \cdot \phi - \rho(\omega) \cdot d\phi = \rho(d \omega) \cdot \phi - \rho(\omega) \cdot d\phi</math>.

Then, according to the example at Template:Section link,

<math>D^2 \phi = \rho(d \omega) \cdot \phi + \rho(\omega) \cdot (\rho(\omega) \cdot \phi) = \rho(d \omega) \cdot \phi + {1 \over 2} \rho([\omega \wedge \omega]) \cdot \phi,</math>

which is <math>\rho(\Omega) \cdot \phi</math> by E. Cartan's structure equation.</ref>

where Template:Nowrap is the representationTemplate:Clarify in <math>\mathfrak{gl}(V)</math> of the curvature two-form Ω. The form F is sometimes referred to as the field strength tensor, in analogy to the role it plays in electromagnetism. Note that D² vanishes for a flat connection (i.e. when Template:Nowrap).

If Template:Nowrap, then one can write

<math>\rho(\Omega) = F = \sum {F^i}_j {e^j}_i</math>

where <math>{e^i}_j</math> is the matrix with 1 at the Template:Nowrap-th entry and zero on the other entries. The matrix <math>{F^i}_j</math> whose entries are 2-forms on P is called the curvature matrix.

For vector bundles

Given a smooth real vector bundle Template:Math with a connection Template:Math and rank Template:Mvar, the exterior covariant derivative is a real-linear map on the vector-valued differential forms that are valued in Template:Mvar:

<math>d^\nabla:\Omega^k(M,E)\to\Omega^{k+1}(M,E).</math>

The covariant derivative is such a map for Template:Math. The exterior covariant derivatives extends this map to general Template:Mvar. There are several equivalent ways to define this object:

• Template:Sfnm Suppose that a vector-valued differential 2-form is regarded as assigning to each Template:Mvar a multilinear map Template:Math which is completely anti-symmetric. Then the exterior covariant derivative Template:Math assigns to each Template:Mvar a multilinear map Template:Math given by the formula

<math>\begin{align}\nabla_{x_1}(s(X_2,X_3))&-\nabla_{x_2}(s(X_1,X_3))+\nabla_{x_3}(s(X_1,X_2))\\ &-s([X_1,X_2],x_3)+s([X_1,X_3],x_2)-s([X_2,X_3],x_1).\end{align}</math>

where Template:Math are arbitrary tangent vectors at Template:Mvar which are extended to smooth locally-defined vector fields Template:Math. The legitimacy of this definition depends on the fact that the above expression depends only on Template:Math, 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. The pattern established in the above formula in the case Template:Math can be directly extended to define the exterior covariant derivative for arbitrary Template:Mvar.

• Template:Sfnm The exterior covariant derivative may be characterized by the axiomatic property of defining for each Template:Mvar a real-linear map Template:Math which for Template:Math is the covariant derivative and in general satisfies the Leibniz rule

<math>d^\nabla(\omega \wedge s) = (d\omega) \wedge s + (-1)^k \omega \wedge (d^\nabla s)</math>

for any differential Template:Mvar-form Template:Math and any vector-valued form Template:Mvar. This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential 1-form Template:Mvar and any local frame Template:Math of the vector bundle, the coordinates of Template:Mvar are locally-defined differential 1-forms Template:Math. The above inductive formula then says thatTemplate:Sfnm

<math>\begin{align}

d^\nabla s&=d^\nabla(\omega^1 e_1+\cdots+\omega^r e_r)\\ &=(d\omega^1) e_1+\cdots+(d\omega^r) e_r-\omega^1 \nabla e_1-\cdots-\omega^r\nabla e_r.\end{align}</math>

In order for this to be a legitimate definition of Template:Math, 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 provides the change-of-basis matrix for the 1-forms Template:Math. When substituted into the above formula, the Leibniz rule as applied for the standard exterior derivative and for the covariant derivative Template:Math cancel out the arbitrary choice.

• Template:Sfnm A vector-valued differential 2-form Template:Mvar may be regarded as a certain collection of functions Template:Math assigned to an arbitrary local frame of Template:Mvar over a local coordinate chart of Template:Mvar. The exterior covariant derivative is then defined as being given by the functions

<math>(d^\nabla s)^\alpha{}_{ijk}=\nabla_is^\alpha{}_{jk}-\nabla_js^\alpha{}_{ik}+\nabla_ks^\alpha{}_{ij}.</math>

The fact that this defines a tensor field valued in Template:Mvar is a direct consequence of the same fact for the covariant derivative. The further fact that it is a differential 3-form valued in Template:Mvar asserts the full anti-symmetry in Template:Math and is directly verified from the above formula and the contextual assumption that Template:Mvar is a vector-valued differential 2-form, so that Template:Math. The pattern in this definition of the exterior covariant derivative for Template:Math can be directly extended to larger values of Template:Mvar.
This definition may alternatively be expressed in terms of an arbitrary local frame of Template:Mvar but without considering coordinates on Template:Mvar. Then a vector-valued differential 2-form is expressed by differential 2-forms Template:Math and the connection is expressed by the connection 1-forms, a skew-symmetric Template:Math matrix of differential 1-forms Template:Math. The exterior covariant derivative of Template:Mvar, as a vector-valued differential 3-form, is expressed relative to the local frame by Template:Mvar many differential 3-forms, defined by

<math>(d^\nabla s)^\alpha=d(s^\alpha)+\theta_\beta{}^\alpha\wedge s^\beta.</math>

In the case of the trivial real line bundle Template:Math 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.

Given a principal bundle, any linear representation of the structure group defines an associated bundle, 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.Template:Sfnm

The curvature of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives Template:Math and Template:Math, so that it is defined as a real-linear map Template:Math. It is a fundamental but not immediately apparent fact that Template:Math only depends on Template:Math, and does so linearly. As such, the curvature may be regarded as an element of Template:Math. 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: Template:Math. In the present context, this can be regarded as saying that the standard connection on the trivial line bundle Template:Math has zero curvature.

Example

• Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero (that is, Template:Nowrap) can be stated as: <math>d\Omega + \operatorname{ad}(\omega) \cdot \Omega = d\Omega + [\omega \wedge \Omega] = 0</math>.

References

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