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Hence, X satisfies NCT, P = Z 1 (M ; C) and eL (h ∗ P)(η ) = 1. 1(iv), the completion of the unstable manifolds are compact; hence, R = I = Z 1 (M ; C). It is well known that Σ = ∅, that is, the integration induces an isomorphism for all η. 1) for all η ∈ Z 1 (M ; R). 1) continues to hold for all η ∈ Z 1 (M ; C). 29 is true for vector fields of the form X = − gradg 0 f . 3. An anomaly formula Consider the bordism W := M × [−1, 1]. Set ∂± W := M × {±1}. Let Y be a vector field on W . Assume that there are vector fields X± on M so that Y (z, s) = X+ (z) + (s − 1)∂/∂s in a neighborhood of ∂+ W and so that Y (z, s) = X− (z) + (−s − 1)∂/∂s in a neighborhood of ∂− W .
Marcsik, ‘Analytic torsion and closed one forms’, PhD Thesis, Ohio State University, Columbus, OH, 1998. 15. V. Mathai and D. Quillen, ‘Superconnections, Thom classes, and equivariant differential forms’, Topology 25 (1986) 85–110. 16. J. Milnor, ‘Whitehead torsion’, Bull. Amer. Math. Soc. 72 (1966) 358–426. 17. S. P. Novikov, ‘Quasiperiodic structures in topology’, Topological methods in modern mathematics, Proceedings of the Symposium in Honor of John Milnor’s Sixtieth Birthday, Stony Brook, NY, 1991 (ed.
Then, there exists ρ > > 0 and a smooth function G : Dρ,ρ → R which is strictly decreasing along nonconstant trajectories of X, which coincides with F on a neighborhood of ∂Dρ,ρ and which coincides with − 12 |y|2 + 12 |z|2 on D , . Proof. Consider the partially defined function which coincides with F in a neighborhood of ∂Dρ,ρ and which coincides with − 21 |y|2 + 12 |z|2 on a neighborhood of D , . We will extend this to a globally defined smooth function Dρ,ρ → R which is strictly decreasing along nonconstant trajectories of X.