Vector-valued differential form

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 differential forms can be viewed as R-valued differential forms. Vector-valued forms are natural objects in differential geometry and have numerous applications.

Formal definition

Let "M" be a smooth manifold and "E" → "M" be a smooth vector bundle over "M". We denote the space of smooth sections of a bundle "E" by Γ("E"). A "E"-valued differential form of degree "p" is a smooth section of the tensor product of "E" with Λ^"p"("T"*"M"), the "p"-th exterior power of the cotangent bundle of "M". The space of such forms is denoted by:$Omega^p(M,E)\; =\; Gamma(EotimesLambda^pT^*M).$By convention, an "E"-valued 0-form is just a section of the bundle "E". That is,:$Omega^0(M,E)\; =\; Gamma(E).,$Equivalently, a "E"-valued differential form can be defined as a bundle morphism:$TMotimescdotsotimes\; TM\; o\; E$which is totally skew-symmetric.

Let "V" be a fixed vector space. A "V"-valued differential form of degree "p" is a differential form of degree "p" with values in the trivial bundle "M" × "V". The space of such forms is denoted Ω^"p"("M", "V"). When "V" = R one recovers the definition of an ordinary differential form.

Operations on vector-valued forms

Pullback

One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an "E"-valued form on "N" by a smooth map φ : "M" → "N" is an (φ*"E")-valued form on "M", where φ*"E" is the pullback bundle of "E" by φ.

The formula is given just as in the ordinary case. For any "E"-valued "p"-form ω on "N" the pullback φ*ω is given by:$(varphi^*omega)\_x(v\_1,cdots,\; v\_p)\; =\; omega\_\{varphi(x)\}(mathrm\; dvarphi\_x(v\_1),cdots,mathrm\; dvarphi\_x(v\_p)).$

Wedge product

Just as for ordinary differential forms, one can define a wedge product of vector-valued forms. The wedge product of a "E"₁-valued "p"-form with a "E"₂-valued "q"-form is naturally a ("E"₁unicode|⊗"E"₂)-valued ("p"+"q")-form::$wedge\; :\; Omega^p(M,E\_1)\; imes\; Omega^q(M,E\_2)\; o\; Omega^\{p+q\}(M,E\_1otimes\; E\_2).$The definition is just as for ordinary forms with the exception that real multiplication is replaced with the tensor product::$(omegawedgeeta)(v\_1,cdots,v\_\{p+q\})\; =\; frac\{1\}\{p!q!\}sum\_\{piin\; S\_\{p+qsgn(pi)omega(v\_\{pi(1)\},cdots,v\_\{pi(p)\})otimes\; eta(v\_\{pi(p+1)\},cdots,v\_\{pi(p+q)\}).$In particular, the wedge product of an ordinary (R-valued) "p"-form with an "E"-valued "q"-form is naturally an "E"-valued ("p"+"q")-form (since the tensor product of "E" with the trivial bundle "M" × R is naturally isomorphic to "E"). For ω ∈ Ω^"p"("M") and η ∈ Ω^"q"("M", "E") one has the usual commutativity relation::$omegawedgeeta\; =\; (-1)^\{pq\}etawedgeomega.$

In general, the wedge product of two "E"-valued forms is "not" another "E"-valued form, but rather an ("E"unicode|⊗"E")-valued form. However, if "E" is an algebra bundle (i.e. a bundle of algebras rather than just vector spaces) one can compose with multiplication in "E" to obtain an "E"-valued form. If "E" is a bundle of commutative, associative algebras then, with this modified wedge product, the set of all "E"-valued differential forms:becomes a graded-commutative associative algebra. If the fibers of "E" are not commutative then Ω("M","E") will not be graded-commutative.

Exterior derivative

For any vector space "V" there is a natural exterior derivative on the space of "V"-valued forms. This is just the ordinary exterior derivative acting component-wise relative to any basis of "V". Explicitly, if {"e"_α} is a basis for "V" then the differential of a "V"-valued "p"-form ω = ω^α"e"_α is given by:$domega\; =\; (domega^alpha)e\_alpha.,$The exterior derivative on "V"-valued forms is completely characterized by the usual relations::More generally, the above remarks apply to "E"-valued forms where "E" is any flat vector bundle over "M" (i.e. a vector bundle whose transition functions are constant). The exterior derivative is defined as above is any local trivialization of "E".

If "E" is not flat then there is no natural notion of an exterior derivative acting on "E"-valued forms. What is needed is a choice of connection on "E". A connection on "E" is a linear differential operator taking sections of "E" to "E"-valued one forms::$abla\; :\; Omega^0(M,E)\; o\; Omega^1(M,E).$If "E" is equipped with a connection ∇ then there is a unique covariant exterior derivative:$d\_\; abla:\; Omega^p(M,E)\; o\; Omega^\{p+1\}(M,E)$extending ∇. The covariant exterior derivative is characterized by linearity and the equation:$d\_\; abla(omegawedgeeta)\; =\; d\_\; ablaomegawedgeeta\; +\; (-1)^p,omegawedge\; deta$where ω is a "E"-valued "p"-form and η is an ordinary "q"-form. In general, one need not have "d"_∇² = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing curvature).

Lie algebra-valued forms

An important case of vector-valued differential forms are Lie algebra-valued forms. These are $mathfrak\; g$-valued forms where $mathfrak\; g$ is a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation is usually denoted [ω∧η] to indicate both operations involved. For example, if ω and η are Lie algebra-valued one forms, then one has:$[omegawedgeeta]\; (v\_1,v\_2)\; =\; [omega(v\_1),eta(v\_2)]\; -\; [omega(v\_2),eta(v\_1)]\; .$With this operation the set of all Lie algebra-valued forms on a manifold "M" becomes a graded Lie superalgebra.

Basic or tensorial forms on principal bundles

Let "E" → "M" be a smooth vector bundle of rank "k" over "M" and let "π" : F("E") → "M" be the (associated) frame bundle of "E", which is a principal GL_"k"(R) bundle over "M". The pullback of "E" by "π" is isomorphic to the trivial bundle F("E") × R^"k". Therefore, the pullback by "π" of an "E"-valued form on "M" determines an R^"k"-valued form on F("E"). It is not hard to check that this pulled back form is right-equivariant with respect to the natural action of GL_"k"(R) on F("E") × R^"k" and vanishes on vertical vectors (tangent vectors to F("E") which lie in the kernel of d"π"). Such vector-valued forms on F("E") are important enough to warrant special terminology: they are called "basic" or "tensorial forms" on F("E").

Let "π" : "P" → "M" be a (smooth) principal "G"-bundle and let "V" be a fixed vector space together with a representation "ρ" : "G" → GL("V"). A basic or tensorial form on "P" of type ρ is a "V"-valued form ω on "P" which is equivariant and horizontal in the sense that
#$(R\_g)^*omega\; =\; ho(g^\{-1\})omega,$ for all "g" ∈ "G", and
#$omega(v\_1,\; ldots,\; v\_p)\; =\; 0$ whenever at least one of the "v"_"i" are vertical (i.e., d"π"("v"_"i") = 0).Here "R"_"g" denotes right-translation by "g" ∈ "G". Note that for 0-forms the second condition is vacuously true.

Given "P" and "ρ" as above one can construct the associated vector bundle "E" = "P" ×_"ρ" "V". Tensorial forms on "P" are in one-to-one correspondence with "E"-valued forms on "M". As in the case of the principal bundle F("E") above, "E"-valued forms on "M" pull back to "V"-valued forms on "P". These are precisely the basic or tensorial forms on "P" of type "ρ". Conversely given any tensorial form on "P" of type "ρ" one can construct the associated "E"-valued form on "M" in a straightforward manner.