Abstract
A heat flow approach to the Godbillon-Vey class
Highlights
Let (M, g) be a compact Riemannian manifold with a codimension 1 foliation F defined by an integrable 1-form ω on M, this is ker(ω) = T F
In particular we prove that if (M, g) is a compact Riemannian manifold with a codimension 1 foliation F, defined by an integrable 1-form ω such that ||ω|| = 1, the Godbillon-Vey class can be written as [−Aω ∧ dω]dR for an operator A : Ω∗(M ) → Ω∗(M ) induced by the heat flow
The integrability of ω guarantees the existence of a 1-form η such that dω = η ∧ ω
Summary
We give a heat flow derivation for the Godbillon Vey class. In particular we prove that if (M, g) is a compact Riemannian manifold with a codimension 1 foliation F , defined by an integrable 1-form ω such that ||ω|| = 1, the Godbillon-Vey class can be written as [−Aω ∧ dω]dR for an operator A : Ω∗(M ) → Ω∗(M ) induced by the heat flow. Defined by an integrable 1-form ω on M , this is ker(ω) = T F . [4] Godbillon and Vey proved that the 3-form η ∧ dη defines a cohomology class gv(F ) ∈ Hd3R(M ) that depends only on F .
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