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The Dirac operator on nilmanifolds and collapsing circle bundles. (English) Zbl 0911.58037

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.

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
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