A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H.

By Veblen O., Whitehead J. H.

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30 For any closed 1-form β on M , the time-1 map f = f1 of the flow of Xβ equals fβ . Proof. 29, the assertion will follow provided that we can show that f ∗ αM = αM + π ∗ β. To this end, note that the definition of the Lie derivative shows that f satisfies 1 ∗ f αM = αM + 0 d ∗ (f αM ) dt = αM + dt t 1 ft∗ (LXβ αM ) dt. 0 By Cartan’s formula for the Lie derivative, we have LXβ αM = d(Xβ αM ) − Xβ ωM = π ∗ β, the latter equality following from the fact that Xβ ⊂ V M ⊂ ker αM and dαM = −ωM . Another application of Cartan’s formula, combined with the assumption that β is closed shows that LXβ π ∗ β = 0, and so ft∗ π ∗ β = π ∗ β for all t.

2. Composition: If pB : B → M is a second submersion and g : B → B is a fiberpreserving diffeomorphism, then (B , pB , φ ◦ g) ∈ M(L, ι, p). 3. Suspension: The suspension of (B, pB , φ) by a nondegenerate quadratic form Q on Rn is defined as the Morse family comprised of the submersion p˜B : B × Rn → M given by composing pB with the projection along Rn , together with the phase function φ˜ = φ + Q. Evidently the fiber-critical set of φ˜ equals the product Σφ × {0}, and ˜ ∈ M(L, ι, p). λφ˜(b, 0) = λφ (b) for all (b, 0) ∈ Σφ˜.

In fact, we will see later that this is the wrong choice! Since exactness is only used to insure that the function eiφ/ is well-defined on L, we can treat certain non-exact cases in a similar way. 1. 4 An immersed lagrangian submanfold (L, ι) ⊂ T ∗ M is said to be prequantizable if its Liouville class λL,ι is -integral for some ∈ R+ . The values of for which this condition holds will again be called admissible for (L, ι). If is admissible for some prequantizable lagrangian immersion (L, ι), then there exists a good cover {Vj } of the manifold L and functions φj : Vj → R such that dφj = ι∗ αM |Vj and φj − φk ∈ Z on each Vj ∩ Vk .

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