2009/08/31 by Ken Richardson, Richardson, Ken
Mathematics · #53C12 #58J50 #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0908.4554
openalex publication_date 2009/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-like metric. We explain the result by Habib-R. that the spectrum of such an operator is independent of the choice of bundle-like metric, provided that the transverse geometric structure is fixed. We discuss consequences, which include defining a new version of the exterior derivative and de Rham cohomology that are nicely adapted to this transverse geometric setting.