Geometry on the Quantum Heisenberg Manifold

Abstract

For the quantum Heisenberg manifolds, using the action of Heisenberg group we construct a family of spectral triples. It is shown that associated Kasparov module is same for all these spectral triples. Then we show that element is nontrivial by explicitly computing the pairing with a unitary. Space of forms as introduced by Connes are also computed. We characterize connections that are torsionless or unitary in the sense of Frohlich et.al. Then it is shown that a torsionless unitary connection cannot exist. For some unitary connections we compute Ricci curvature and scalar curvature. This produces examples of non-commutative manifolds with nonvanishing scalar curvature.