Infinitesimal deformations of foliations and Cartan connections
Taro Asuke

In [1] and [2], Heitsch gave a construction of derivatives of secondary characteristic classes with respect to infinitesimal deformations of foliations. Indeed, if there is a smooth family {Ft} of codimension-q foliations on a manifold M and if ω is an element of Hp(WOq)(or Hp(Wq), Hp(WUq) and so on), then (d/dt)ω(Ft) is defined as an element of Hp(M). Such a family determines an element, say σ(Ft), of H1(M;ΘF0) (essentially by the Kodaira-Spencer theory). On the other hand, if σ ∈ H1(M;ΘF0), then an element Dσω ∈ Hp(M) is defined in a way such that if σ = σ(Ft) then D σ(F t )ω = (d/dt)ω(Ft)|t=0. The original construction makes use of a certain resolution of ΘF0 . In this talk, I will explain a reconstruction of the derivatives in terms of Cartan connections and jet bundles.
[1] J. Heitsch, A cohomology for foliated manifolds, Comment. Math. Helv. 15 (1975), 197–218.
[2] J. Heitsch, Derivatives of secondary characteristic classes, J. Differential Geometry 13 (1978),
2000 Mathematics Subject Classification. Primary 57R30; Secondary 58H05, 37F35, 37F75. 1
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