Math 396. From integral curves to integral manifolds 1. Integral manifolds for trivial line bundles

Let M be a C∞ manifold (without corners) and let E ⊆ TM be a subbundle of the tangent bundle. In class we discussed the notion of integral manifolds for E in M (as well as maximal ones), essentially as a generalization of the theory of integral curves for vector fields. Roughly speaking, 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 changing the trivialization. Say ~v and ~ w are two trivializations for a line subbundle L in TM , which is to say 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 onM . We shall prove that the associated maximal integral curves are “the same” up to a unique reparameterization in time that fixes t = 0: