Stanford MATH 396 - Integral manifolds for trivial line bundles

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Math 396. From integral curves to integral manifolds1. Integral manifolds for trivial line bundlesLet M be a C∞manifold (without corners) and let E ⊆ T M be a subbundle of the tangentbundle. In class we discussed the notion of integral manifolds for E in M (as well as maximalones), essentially as a generalization of the theory of integral curves for vector fields. Roughlyspeaking, in the special case that E is a trivial line bundle we are in the setup of integral curves for(non-vanishing) vector fields but with a fundamental difference: we do not specify the trivialization.What is the impact of this?To motivate what is to follow, we shall now undertake a close study of the effect of changingthe trivialization. Say ~v and ~w are two trivializations for a line subbundle L in T M, which is tosay that these are non-vanishing smooth vector fields which are pointwise proportional, so we have~w = f~v for a necessarily non-vanishing smooth function f on M. We shall prove that the associatedmaximal integral curves are “the same” up to a unique reparameterization in time that fixes t = 0:Theorem 1.1. The respective maximal integral curves c : I → M and ec : J → M for ~v and ~wthrough m0at time 0 satisfy ec = c ◦ F for a unique C∞isomorphism F : J ' I preserving 0. Inparticular, c(I) = ec(J) as subsets of M.If you visualize the meaning of this theorem in terms of motion of two particles (parameterizedby time), the assertion becomes “physically obvious”. The rigorous proof thereby illustrates thedifference between physics and mathematics. Keep the visualization in mind when reading theproof, since it both motivates the entire strategy of proof and makings it easy to understand.Proof. By definition, c0(t) = ~v(c(t)) for all t ∈ I with c(0) = m0, and I is the unique maximal openinterval in R with this property (it contains all others). Since f ◦ c is a non-vanishing continuousfunction on the open interval I, it has constant sign. If f is negative then we can replace ~v with −~v,I with −I, and t 7→ c(t) on I with t 7→ c(−t) on −I without changing the image of c but bringingus to the case f > 0. Hence, we now suppose f > 0. Consider the I-valued initial-value problemF0= (f ◦ c) ◦ F with initial condition F (0) = 0 ∈ int(I) for smooth maps F : J0→ I on intervalsJ0around 0. This is a non-linear ODE, and the local existence theorem ensures that there existssuch a solution on some interval J0around 0. Since f is positive, so is F0, and hence F is a strictlyincreasing function. Thus, F is a C∞order-preserving isomorphism of J0onto F (J0).The old arguments via uniqueness of solutions to ODE’s provide a maximal open interval J0onwhich there is a solution (satisfying F (J0) ⊆ I!), and it is unique. Taking J0to be maximal, wemust have F (J0) = I: if not then the strictly increasing F is bounded away from endpoints of I ast appoaches some endpoint of J0yet the smooth f ◦ c persists across all of I and so by Corollary2.5 in the handout on ODE’s (!) the interval J0would not be maximal after all. The smooth mapc ◦ F : J0→ M carries 0 to m0and satisfies(c ◦ F )0(t) = F0(t) · c0(F (t)) = (f ◦ c)(F (t)) · ~v(c(F (t))) = f((c ◦ F )(t)) ·~v((c ◦ F )(t)) = ~w((c ◦ F )(t)).This says that c ◦ F : J0→ M is an integral curve for ~w through m0at time 0. In particular, J0is contained in the open interval of definition J for the maximal integral curve ec : J → M for ~wpassing through m0at time 0.Note that ~w = (1/f) · ~v. By the formula for the derivative of an inverse function in 1-variablecalculus, the smooth strictly increasing map F−1: I ' J0⊆ J sending 0 to 0 solves the ODEy0= ((1/f ) ◦ ec) ◦ y for J-valued y on I because ec|J0= c ◦ F . By the argument given above, if weconsider the solution H :eI → J to this initial-value problem on a maximal open interval of definitionaround 0 (so I ⊆eI and H|eI= F−1) then H(eI) = J and ec ◦ H :eI → M is an integral curve for ~v12through m0at time 0. This forceseI ⊆ I, so in facteI = I and J = H(eI) = H(I) = F−1(I) = J0.We have therefore proved that J0= J, so F is a strictly increasing C∞isomorphism between themaximal intervals of definition for the integral curves of ~v and ~w = f~v through m0at time 0, andcomposition with F carries the maximal integral curve for ~v to the maximal integral curve for ~w.It remains to check that F is the unique solution to our problem: if H : J ' I is a C∞isomorphism fixing the origin such that ec = c ◦ H, then H = F . By differentiating and using the“integral curve” properties of c and ec,f(ec(t)) · ~v(ec(t)) = ~w(ec(t)) = H0(t) · ~v(c(H(t))) = H0(t) · ~v(ec(t)).Since ~v(ec(t)) 6= 0 for all t, we get H0(t) = f (ec(t)) = (f ◦ c)(H(t)). Hence H : J → I is a solutionto the same initial-value problem as F (i.e., y0(t) = (f ◦ c)(y (t)) for I-valued y on an open intervalaround 0, with y(0) = 0). This forces H = F . We have just shown that if we replace a non-vanishing vector field ~v on M with the line subbundleL ⊆ T M that it generates (this is just the C∞subbundle inclusion M × R → T M given by(m, a) 7→ a~v(m)), then since L only “knows” ~v up to multiplication by a non-vanishing smoothfunction it only “knows” the maximal integral curve c~v,m0: I~v,m0→ M up to (possibly order-reversing) composition with some C∞isomorphism F : J ' I~v,m0for an open subinterval J of Raround 0. In particular, the image subset c~v,m0(I~v,m0) and the property of whether or not c~v,m0is injective depend only on m0and not on the choice of trivialization ~v for L. Let us write Nm0for this image subset. In the non-injective case, we know from Example 5.7 in the handout onintegral curves that Nm0is a smoothly embedded circle in M. As we have seen long ago for generalembedded submanifolds of a manifold, this is the unique possible C∞submanifold structure on thesubset Nm0in M in such cases. In the injective case, the non-vanishing of the vector field impliesthat c~v,m0: I~v,m0→ M is an injective immersion, though this may not be an embedding (thinkof the line densely wrapping the torus). We thereby get a (possibly non-embedded) submanifoldstructure on the subset Nm0⊆ M by using the bijection c~v,m0and the C∞manifold structureon I~v,m0to put both the topology and differentiable structure on Nm0. If we change ~v to someother ~w trivializing L then the preceding arguments show that there is a (unique) C∞isomorphismF : I~w,m0'


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Stanford MATH 396 - Integral manifolds for trivial line bundles

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