Ammann, Bernd; Bär, Christian The Dirac operator on nilmanifolds and collapsing circle bundles. (English) Zbl 0911.58037 Ann. Global Anal. Geom. 16, No. 3, 221-253 (1998). The spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds is computed and the behaviour under collapse to the 2-torus is studied. The Dirac eigenvalues on complex projective space including the multiplicities are determined by using the Hopf fibration and spin structures. It is shown that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have a constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have a nonconstant spectrum. Reviewer: Sergui I.Văcaru (Chisinau) Cited in 2 ReviewsCited in 26 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds Keywords:circle bundles; collapse; Dirac operator; Heisenberg manifolds; isospectral deformation; nilmanifolds; spin structures; Laplacian PDFBibTeX XMLCite \textit{B. Ammann} and \textit{C. Bär}, Ann. Global Anal. Geom. 16, No. 3, 221--253 (1998; Zbl 0911.58037) Full Text: DOI arXiv