# The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

by

Bernd Ammann and Christian Bär

**The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles**
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*Ann. Global Anal. Geom.* **16**, no. 3,
*221-253, 1998*

We compute the spectrum of the Dirac operator on 3-dimensional
Heisenberg manifolds.
The behavior under collapse to the 2-torus is studied.
Depending on the spin structure either all eigenvalues tend to
$\pm\infty$ or there are eigenvalues converging to those of the
torus.
This is shown to be true in general for collapsing circle bundles
with totally geodesic fibers.
Using the Hopf fibration we use this fact to compute the Dirac
eigenvalues on complex projective space including the multiplicities.
Finally, we show that there are 1-parameter families of Riemannian
nilmanifolds such that the Laplacian on functions and the Dirac
operator for certain spin structures have constant spectrum while
the Laplacian on 1-forms and the Dirac operator for the other
spin structures have nonconstant spectrum.
The marked length spectrum is also constant for these families.
58G25, 58G30, 53C25, 53C30
### Keywords

Dirac operator, nilmanifolds, Heisenberg manifolds, circle bundles,
collapse, isospectral deformation

Bernd Ammann, 30.9.1998