"Tangential Chern-Weil homomorphism"

GEOMETRIC STUDY OF FOLIATIONS, World Scientific, Singapure, 1994, 327-344; Proceedings of the International Symposium/Workshop, 15-26 Nov 1993, Tokyo, Japan.
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Moore and Schochet gave the Chern-Weil homomorphism of a vector bundle f over a foliated space (M, F), measuring the nonexistence of partially flat covariant derivatives. We look at this problem (restricting our interest to vector bundles over foliated manifolds) from the point of view of nontransitive Lie algebroids. We use – for our problem – the Chern-Weil homomorphism of the regular Lie algebroid coming from the Lie algebroid A(f) of f by restricting it to the elements whose anchors are tangent to the foliation F. Our observations lead to the conjecture that we can sometimes obtain some essentially new kinds of characteristic classes (with respect to the construction of Moore-Schochet) called singular.