Betti numbers, characteristic classes and sub-Riemannian geometry
type: Item
instance of: scholarly article
title: Betti numbers, characteristic classes and sub-Riemannian geometry
zbMATH Open document ID: 0769.53014
author: Zhong Ge
published in: Illinois Journal of Mathematics
publication date: 1993-02-10T00:00:00Z
review text: 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 \(P\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58J10
zbMATH DE Number: 118439
zbMATH Keywords: horizontal connections
zbMATH Keywords: Hörmander's condition
zbMATH Keywords: horizontal cohomology
zbMATH Keywords: horizontal Laplacian
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 118439