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
H^{1}(M;Θ_{F0}) (essentially by the Kodaira-Spencer theory). On the other hand, if σ ∈ H^{1}(M;Θ_{F0}), then an element D_{σ}ω ∈ 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