Betti numbers, characteristic classes and sub-Riemannian geometry
Let \(H\) be a smooth subbundle of the tangent bundle of a manifold \(M\) which satisfies Hörmander's condition. The author studies differentiable invariants of \(H\) as follows: 1. A notion of horizontal cohomology of \(H\) is introduced. If \(H\) satisfies the strong bracket generating hypothesis then some of the low dimensional cohomology groups coincide with the usual de Rham cohomology groups of \(M\). 2. The Chern- Weil theory of characteristic classes for a vector bundle \(V\) over \(M\) is generalized to ``horizontal connections'' for \(V\) defined along \(H\). 3. A (non-canonical) horizontal Laplacian of a sub-Riemannian metric on \(H\) is defined and used to investigate principal fibre bundles \(P\) over Lie groups with horizontal bundles \(H\) defined by a connection in \(P\).
scientific article; zbMATH DE number 118439