×

The eta invariant on two-step nilmanifolds. (English) Zbl 1394.58009

The \(\eta\)-invariant of a Dirac operator \(D_M\) on a closed manifold \(M\) was defined by M. F. Atiyah et al. [Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975; Zbl 0297.58008)]. It is very difficult to calculate the eta invariant for a given operator such as a Dirac operator on a Riemannian manifold; much work has been done to calculate this invariant for space forms, lens spaces and tori. In this work, the authors derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, they study the eta invariant on nilmanifolds by decomposing the spin Dirac operator using Kirillov theory. In particular, for general Heisenberg three-manifolds, the spectrum of the Dirac operator and the eta invariant are computed in terms of the metric, lattice, and spin structure data.

MSC:

58J28 Eta-invariants, Chern-Simons invariants
22E25 Nilpotent and solvable Lie groups
35P20 Asymptotic distributions of eigenvalues in context of PDEs
53C27 Spin and Spin\({}^c\) geometry
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 0297.58008
PDFBibTeX XMLCite
Full Text: DOI arXiv