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 H^p(WO_q)(or H^p(W_q), H^p(WU_q) and so on), then (d/dt)ω(Ft) is defined as an element of H^p(M). Such a family determines an element, say σ(Ft), of H¹(M;Θ_F0) (essentially by the Kodaira-Spencer theory). On the other hand, if σ ∈ H¹(M;Θ_F0), then an element D_σω ∈ H^p(M) is defined in a way such that if σ = σ(F_t) then D _{σ(F t )}ω = (d/dt)ω(F_t)|_t=0. The original construction makes use of a certain resolution of Θ_F0 . In this talk, I will explain a reconstruction of the derivatives in terms of Cartan connections and jet bundles.
References
[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
E-mail address: asuke@ms.u-tokyo.ac.jp