Editing Exterior covariant derivative

{{see also|Second covariant derivative}}
In the [[mathematics|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 (principal bundle)|connection]].

==Definition==
Let ''G'' be a [[Lie group]] and {{nowrap|''P'' → ''M''}} be a [[principal bundle|principal ''G''-bundle]] on a [[smooth manifold]] ''M''. Suppose there is a [[connection (principal bundle)|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 subspace|horizontal]] and [[vertical subspace|vertical]] subspaces. Let <math>h: T_u P \to H_u</math> be the projection to the horizontal subspace.

If ''ϕ'' is a [[vector-valued form|''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''<sub>''i''</sub> are tangent vectors to ''P'' at ''u''.

Suppose that {{nowrap|''ρ'' : ''G'' → GL(''V'')}} is a [[representation of a Lie group|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 {{nowrap|(''k'' + 1)}}-form]] on ''P'' of the type ''ρ'': it is equivariant and horizontal (a form ''ψ'' is horizontal if {{nowrap|1=''ψ''(''v''<sub>0</sub>, ..., ''v''<sub>k</sub>) = ''ψ''(''hv''<sub>0</sub>, ..., ''hv''<sub>''k''</sub>)}}.)

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 [[Lie algebra-valued differential form#operations|<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 {{nowrap|1=''k'' = 0}}, 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 {{nowrap|1=''ϕ''(''gu'') = ''ρ''(''g''<sup>−1</sup>)''ϕ''(''u'')}}. On the other hand, {{nowrap|1=''Dϕ''(''X''<sup>#</sup>) = 0}}. 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''<sub>''i''</sub>'s be tangent vectors to ''P'' at some point such that some of ''X''<sub>''i''</sub>'s are horizontal and the rest vertical. If ''X''<sub>''i''</sub> 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''<sub>''i''</sub> is horizontal, we replace it with the [[horizontal lift]] of the vector field extending the pushforward π''X''<sub>''i''</sub>. This way, we have extended ''X''<sub>''i''</sub>'s to vector fields. Note the extension is such that we have: [''X''<sub>''i''</sub>, ''X''<sub>''j''</sub>] = 0 if ''X''<sub>''i''</sub> is horizontal and ''X''<sub>''j''</sub> 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 ''{{section link|Lie algebra-valued differential form#Operations}}'', 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>

<!--
If one chooses a basis ''e''<sub>''i''</sub> of <math>\mathfrak{g}</math>, then the formula reads:
:<math>(D \phi)^k = d \phi^k + \rho_{ij}^k \omega^i \wedge \phi^j</math>
where <math>\phi = \sum \phi^i e_i</math>, the same for ''ω'' and ''Dϕ'' and <math>\rho(e_i)e_j = \sum \rho_{ij}^k e_k</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 {{section link|Lie algebra-valued differential form|Operations}},
:<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 {{nowrap|1=''F'' = ''ρ''(Ω)}} is the representation{{clarify|what does "representation" mean here?|date=December 2018}} in <math>\mathfrak{gl}(V)</math> of the [[curvature form|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 ''D''<sup>2</sup> vanishes for a [[flat connection]] (i.e. when {{nowrap|1=Ω = 0}}).

If {{nowrap|''ρ'' : ''G'' → GL('''''R'''''<sup>''n''</sup>)}}, 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 {{nowrap|(''i'', ''j'')}}-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 {{math|''E'' → ''M''}} with a [[connection (vector bundle)|connection]] {{math|∇}} and rank {{mvar|r}}, the '''exterior covariant derivative''' is a real-linear map on the [[vector-valued differential forms]] that are valued in {{mvar|E}}:
:<math>d^\nabla:\Omega^k(M,E)\to\Omega^{k+1}(M,E).</math>
The covariant derivative is such a map for {{math|''k'' {{=}} 0}}. The exterior covariant derivatives extends this map to general {{mvar|k}}. There are several equivalent ways to define this object:
* {{sfnm|1a1=Besse|1y=1987|1loc=Section 1.12|2a1=Kolář|2a2=Michor|2a3=Slovák|2y=1993|2loc=Section 11.13}} Suppose that a vector-valued differential 2-form is regarded as assigning to each {{mvar|p}} a multilinear map {{math|''s''<sub>''p''</sub>: ''T''<sub>''p''</sub>''M'' × ''T''<sub>''p''</sub>''M'' → ''E''<sub>''p''</sub>}} which is completely anti-symmetric. Then the exterior covariant derivative {{math|d<sup>∇</sup> ''s''}} assigns to each {{mvar|p}} a multilinear map {{math|''T''<sub>''p''</sub>''M'' × ''T''<sub>''p''</sub>''M'' × ''T''<sub>''p''</sub>''M'' → ''E''<sub>''p''</sub>}} 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 {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>}} are arbitrary tangent vectors at {{mvar|p}} which are extended to smooth locally-defined vector fields {{math|''X''<sub>1</sub>, ''X''<sub>2</sub> ''X''<sub>3</sub>}}. The legitimacy of this definition depends on the fact that the above expression depends only on {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>}}, 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 {{math|''k'' {{=}} 2}} can be directly extended to define the exterior covariant derivative for arbitrary {{mvar|k}}.
* {{sfnm|1a1=Donaldson|1a2=Kronheimer|1y=1990|1p=35|2a1=Eguchi|2a2=Gilkey|2a3=Hanson|2y=1980|2p=281|3a1=Jost|3y=2017|3p=169|4a1=Taylor|4y=2011|4p=547}} The exterior covariant derivative may be characterized by the axiomatic property of defining for each {{mvar|k}} a real-linear map {{math|&Omega;<sup>''k''</sup>(''M'', ''E'') → &Omega;<sup>''k'' + 1</sup>(''M'', ''E'')}} which for {{math|''k'' {{=}} 0}} 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 {{mvar|k}}-form {{math|&omega;}} and any vector-valued form {{mvar|s}}. This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential 1-form {{mvar|s}} and any local frame {{math|''e''<sub>1</sub>, ..., ''e''<sub>''r''</sub>}} of the vector bundle, the coordinates of {{mvar|s}} are locally-defined differential 1-forms {{math|''&omega;''<sup>1</sup>, ..., ''&omega;''<sup>''r''</sup>}}. The above inductive formula then says that{{sfnm|1a1=Milnor|1a2=Stasheff|1y=1974|1pp=292–293}}
::<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 {{math|''d''<sup>∇</sup>''s''}}, 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 {{math|''&omega;''<sub>1</sub>, ..., ''&omega;''<sub>''r''</sub>}}. When substituted into the above formula, the Leibniz rule as applied for the standard exterior derivative and for the covariant derivative {{math|∇}} cancel out the arbitrary choice.
* {{sfnm|1a1=Eells|1a2=Sampson|1y=1964|1loc=Section 3.A.3|2a1=Penrose|2a2=Rindler|2y=1987|2p=263}} A vector-valued differential 2-form {{mvar|s}} may be regarded as a certain collection of functions {{math|''s''<sup>&alpha;</sup><sub>''ij''</sub>}} assigned to an arbitrary local frame of {{mvar|E}} over a local coordinate chart of {{mvar|M}}. 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 {{mvar|E}} is a direct consequence of the same fact for the covariant derivative. The further fact that it is a differential 3-form valued in {{mvar|E}} asserts the full anti-symmetry in {{math|''i'', ''j'', ''k''}} and is directly verified from the above formula and the contextual assumption that {{mvar|s}} is a vector-valued differential 2-form, so that {{math|''s''<sup>&alpha;</sup><sub>''ij''</sub> {{=}} −''s''<sup>&alpha;</sup><sub>''ji''</sub>}}. The pattern in this definition of the exterior covariant derivative for {{math|''k'' {{=}} 2}} can be directly extended to larger values of {{mvar|k}}.<br />This definition may alternatively be expressed in terms of an arbitrary local frame of {{mvar|E}} but without considering coordinates on {{mvar|M}}. Then a vector-valued differential 2-form is expressed by differential 2-forms {{math|''s''<sup>1</sup>, ..., ''s''<sup>''r''</sup>}} and the connection is expressed by the connection 1-forms, a skew-symmetric {{math|''r'' × ''r''}} matrix of differential 1-forms {{math|&theta;<sub>&alpha;</sub><sup>&beta;</sup>}}. The exterior covariant derivative of {{mvar|s}}, as a vector-valued differential 3-form, is expressed relative to the local frame by {{mvar|r}} 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 {{math|ℝ × ''M'' → ''M''}} 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 [[representation of a Lie group|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 [[Vector-valued differential forms#Basic or tensorial forms on principal bundles|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.{{sfnm|1a1=Kolář|1a2=Michor|1a3=Slovák|1y=1993|1pp=112–114}}

The [[Curvature form|curvature]] of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives {{math|&Omega;<sup>0</sup>(''M'', ''E'') → &Omega;<sup>1</sup>(''M'', ''E'')}} and {{math|&Omega;<sup>1</sup>(''M'', ''E'') → &Omega;<sup>2</sup>(''M'', ''E'')}}, so that it is defined as a real-linear map {{math|''F'': &Omega;<sup>0</sup>(''M'', ''E'') → &Omega;<sup>2</sup>(''M'', ''E'')}}. It is a fundamental but not immediately apparent fact that {{math|''F''(''s'')<sub>''p''</sub>: ''T''<sub>''p''</sub>''M'' × ''T''<sub>''p''</sub>''M'' → ''E''<sub>''p''</sub>}} only depends on {{math|''s''(''p'')}}, and does so linearly. As such, the curvature may be regarded as an element of {{math|&Omega;<sup>2</sup>(''M'', End(''E''))}}. 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: {{math|''d''(''d''&omega;) {{=}} 0}}. In the present context, this can be regarded as saying that the standard connection on the trivial line bundle {{math|ℝ × ''M'' → ''M''}} has zero curvature.

== Example ==
* [[Bianchi's second identity]], which says that the exterior covariant derivative of Ω is zero (that is, {{nowrap|1=''D''Ω = 0}}) can be stated as: <math>d\Omega + \operatorname{ad}(\omega) \cdot \Omega = d\Omega + [\omega \wedge \Omega] = 0</math>.

==Notes==
{{reflist}}

==References==
{{refbegin}}
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*{{cite book|last1=Donaldson|first1=S. K.|last2=Kronheimer|first2=P. B.|title=The geometry of four-manifolds|series=Oxford Mathematical Monographs|publisher=[[Oxford University Press|The Clarendon Press, Oxford University Press]]|location=New York|year=1990|isbn=0-19-853553-8|author-link1=Simon Donaldson|author-link2=Peter Kronheimer|mr=1079726|zbl=0820.57002}}
*{{cite journal|last1=Eells|first1=James Jr.|last2=Sampson|first2=J. H.|title=Harmonic mappings of Riemannian manifolds|journal=[[American Journal of Mathematics]]|volume=86|year=1964|pages=109–160|mr=0164306|author-link1=James Eells|author-link2=Joseph H. Sampson|issue=1|doi=10.2307/2373037| jstor=2373037 |zbl=0122.40102}}
*{{cite journal|last1=Eguchi|first1=Tohru|last2=Gilkey|first2=Peter B.|last3=Hanson|first3=Andrew J.|title=Gravitation, gauge theories and differential geometry|journal=[[Physics Reports]]|volume=66|year=1980|issue=6|pages=213–393|mr=0598586|author-link1=Tohru Eguchi|author-link2=Peter Gilkey|author-link3=Andrew Hanson|doi=10.1016/0370-1573(80)90130-1| bibcode=1980PhR....66..213E }}
*{{cite book|last1=Jost|first1=Jürgen|title=Riemannian geometry and geometric analysis| series=Universitext |author-link1=Jürgen Jost|edition=Seventh edition of 1995 original|publisher=[[Springer, Cham]]|year=2017|isbn=978-3-319-61859-3|mr=3726907|doi=10.1007/978-3-319-61860-9|zbl=1380.53001}}
*{{cite book|author-link1=Shoshichi Kobayashi|mr=0152974|zbl=0119.37502|last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi|author-link2=Katsumi Nomizu|title=Foundations of differential geometry. Vol I|title-link=Foundations of differential geometry|publisher=[[John Wiley & Sons, Inc.]]|location=New York–London|year=1963|others=Reprinted in 1996|isbn=0-471-15733-3|series=Interscience Tracts in Pure and Applied Mathematics|volume=15| issue=1 }}
* {{cite book|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter W.|last3=Slovák|first3=Jan|title=Natural operations in differential geometry|publisher=[[Springer-Verlag]]|location=Berlin|year=1993|isbn=3-540-56235-4|mr=1202431|zbl=0782.53013|url=https://www.emis.de///monographs/KSM/}}
*{{cite book|last1=Milnor|first1=John W.|last2=Stasheff|first2=James D.|title=Characteristic classes|series=Annals of Mathematics Studies|volume=76|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1974|mr=0440554|author-link1=John Milnor|author-link2=James Stasheff|zbl=0298.57008|doi=10.1515/9781400881826|isbn=0-691-08122-0}}
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*{{cite book|last1=Penrose|first1=Roger|last2=Rindler|first2=Wolfgang|title=Spinors and space-time. Vol. 1. Two-spinor calculus and relativistic fields|series=Cambridge Monographs on Mathematical Physics|publisher=[[Cambridge University Press]]|location=Cambridge|year=1987|isbn=0-521-33707-0|author-link1=Roger Penrose|author-link2=Wolfgang Rindler|mr=0917488|edition=Second edition of 1984 original|doi=10.1017/CBO9780511564048|zbl=0602.53001}}
*{{cite book|mr=2743652|last1=Taylor|first1=Michael E.|author-link1=Michael E. Taylor|title=Partial differential equations II. Qualitative studies of linear equations|edition=Second edition of 1996 original|series=Applied Mathematical Sciences|volume=116|publisher=[[Springer Publishing|Springer]]|location=New York|year=2011|isbn=978-1-4419-7051-0|doi=10.1007/978-1-4419-7052-7|zbl=1206.35003}}
{{refend}}

{{Tensors}}

[[Category:Connection (mathematics)]]
[[Category:Differential geometry]]
[[Category:Fiber bundles]]