Math 396. Local and global Frobenius theorems1. MotivationLet M be a smooth manifold, and E an integrable subbundle of T M. A particularly interestingexample is the following. Let M = G be a Lie group and h a Lie subalgebra of g = Lie(G). Inthis case, we have a bundle trivialization G × g ' T G given by the construction of left-invariantvector fields. This is the C∞bundle map (g, v) 7→ (dλg(e))(v) that is an isomorphism on fibers(the smoothness was shown in the home work), and in this way G × h inside of G × g goes overto a subbundleeh in T G given by propogating elements of h by left translation. That is, the C∞bundle mapping G × h → T G defined by (g, v) 7→ (dλg(e))(v) is fiberwise injective (it puts h insideof Tg(G) via left translation by g), hence it is a subbundle of T G, and the crux is this: because his a Lie subalgebra (rather than an arbitrary linear subspace of g) this subbundle is integrable. Toprove this fact, first observe that by construction if X ∈ h ⊆ g then the associated left-invariantvector fieldeX on G is a C∞section of the subbundleeh ⊆ T G (why?), and so if X1, . . . , Xnis abasis of h theneX1, . . . ,eXnis clearly a global trivializing frame foreh (why?). In general, to proveintegrability of a subbundle of the tangent bundle it suffices (as we have seen in class) to provethat the bracket operation applied to members of a trivializing frame over the constituents of anopen cove ring of the base space yields output that is a section of the subbundle. In our case thereis the global trivializing frameeX1, . . . ,eXnofeh, so to prove integrability ofeh ⊆ T G we just have toprove that [eXi,eXj] ∈eh(G) inside of (T G)(G) = VecG(G). ButeXiandeXjare left-invariant vectorfields on G, so by the very definition of the Lie algebra structure on g = Te(G) in terms of thecommutator operation on global left-invariant vector fields we have [eXi,eXj] = [Xi, Xj]∼. That is,the bracket ofeXiandeXjis equal to the left-invariant vector field associated to the tangent vector[Xi, Xj] ∈ g. But Xi, Xj∈ h and by hypothesis h is a Lie subalgebra of g. Hence, [Xi, Xj] ∈ h, so byconstruction when this is propogated to a left-invariant vector field on G the resulting global vectorfield is a section of the subbundleeh in T G (by how this subbundle was defined). This concludesthe verification thateh is indeed an integrable subbundle of T G.We shall see later in that handout that a maximal integral submanifold H in G to the integrablesubbundleeh such that H contains the identity is a connected Lie subgroup of G (by which we meanan injective immersion of Lie groups i : H → G that respects the group structures) and that itsassociated Lie subalgebra Lie(H) ⊆ g is the initial choice of Lie subalgebra h.Example 1.1. Let G = GLn(R). The Lie algebra is denoted gln(R), and as a vector space isnaturally identified with the vector space Matn×n(R) (as G is an ope n submanifold of Matn×n(R)).I claim that the Lie algebra structure on gln(R) is thereby identified with the “usual” bracket onn × n matrices, namely [A, B] = AB − BA.How can we prove this? There is a clever way to prove this using some general principles from thetheory of Lie groups, but in the present setting it can be proved rather concretely. Let A = (aij)be an element of Matn×n(R) viewed as gln(R), which is to say (by the realization of “matrixentries” xijas a linear coordinate system on the vector space Matn×n(R) containing G as an opensubmanifold) that A corresponds to the tangent vector~A =Paij∂xij|ein Te(G). By using themethod of solution to Exercise 2(iv) in Homework 7 (i.e., the formula for matrix multiplicationin terms of matrix entries) each ∂xij|eextends to the left-invariant vector fieldPkxki∂xkjon G.Hence, the left-invariant vector field with value~A at the identity is the corresponding R-linear12combinatione~A =Xi,jaijXkxki∂xkj=Xk,j(Xiaijxki) · ∂xkj.This is really to be viewed as a vector field on the open submanifold G ⊆ Matn×n(R), though itmakes perfectly good sense even on Matn×n(R).The calculation of the commutator of global vector fieldse~A ande~B on G is now a matter ofalgebra:Xk,j,k0,j0Xi,i0aijbi0j0[xki∂xkj, xk0i0∂xk0j0] =Xk,j,k0,j0Xi,i0aijbi0j0(δ(k,j),(k0,i0)xki∂xk0j0− δ(k,i),(k0,j0)xk0i0∂xkj)=Xi,j,j0,kaijbjj0xki∂xkj0−Xi,j,i0,kaijbi0ixki0∂xkj.Evaluating at the identity point (xrs) = (δrs), this collapses toXi,j,j0aijbjj0∂xij0|e−Xi,j,kaijbki∂xkj|e=Xr,s(Xmarmbms− brmams)∂xrs|e.The rs-coefficient is the rs-entry of the matrix commutator AB − BA, so passing from this vectorin Te(G) back to the language of Matn×n(R) we obtain the desired description of the Lie algebrastructure on gln(R) as the commutator of n × n matrices.Since we have described the Lie algebra structure on gln(R) = Matn×n(R) as just the ordinarycommutator AB − BA of matrices, one Lie subalgebra jumps out at us: the subspace of trace 0matrices. This is a hyperplane that is stable under the bracket because every bracket in the Liealgebra lies in here (clearly AB − BA has trace 0 for any A and B). This turns out to be thetangent space to the c onnected (!) closed Lie subgroup SLn(R) of matrices with determinant 1.There are lots of other Lie subalgebras of more interesting nature. For example, since [A, B]t=[Bt, At] = −[At, Bt] for matrices A and B, if A and B are skew-symmetric then so is [A, B]. Hence,the subspace son(R) ⊆ gln(R) of skew-symmetric matrices is a Lie subalgebra. This turns out tobe the tangent space to the connected (!) closed Lie subgroup SOn(R) of orthogonal matrices withdeterminant 1.In this handout, we wish to give a general statement of the local and global Frobenius theorems,some discussions concerning the proofs, and work out the general application to the proof of exis-tence and uniqueness of a connected Lie subgroup H of a Lie group G such that Lie(H) ⊆ Lie(G)coincides with a given Lie subalgebra h ⊆ Lie(G).2. Statement of main resultsHere is the local theorem:Theorem 2.1 (Frobenius). Let E be an fiberwise nonzero integrable subbundle of T M, for Ma smooth manifold. There exists a covering of M by C∞charts (U, ϕ) with ϕ = {x1, . . . , xn}a C∞coordinate system with ϕ(U) =Q(ai, bi) ⊆ Rna product of open intervals such that forr = rank(E|U) the embedded r -dimensional slice submanifolds {xi= ci}i>rfor (cr+1, . . . , cn) ∈Qi>r(ai, bi) are integral manifolds for E. Moreover, all (connected!) integral manifolds for