Homotopy and Geometry, Banach Center Publications, Volume 45, Institute of Mathematics, Polish Academy of Sciences, Warszawa 1998, 199-224.
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Two following homotopic notions are important in many domains of differential geometry:
– homotopic homomorphisms between principal bundles (and between other objects),
– homotopic subbundles.
They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be – in a natural way – expressed in the language of Lie algebroids. Besides, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism, or the subject of this paper, the characteristic homomorphism for flat bundles) are invariants of Lie algebroids of these bundles. This enables one to construct the characteristic homomorphism of a flat regular Lie algebroid, measuring the incompatibility of the flat structure with a given subalgebroid. For two given Lie subalgebroids, these homomorphisms are equivalent if the Lie subalgebroids are homotopic. Some new examples of applications of this characteristic homomorphism to a transitive case (for TC-foliations) and to a non-transitive case (for a principal bundle equipped with a partial flat connection) are pointed to (Ex 3.1). An example of a transitive Lie algebroid of a TC-foliation, which leads to the nontrivial characteristic homomorphism is obtained.