Quantum Riemannian Structures

Abstract

This chapter brings together previously encountered notions of Riemannian connections and other structures from Chapters 3 , 4 , 5 into a self-contained account of noncommutative Riemannian geometry over an algebra equipped with differential structure and choice of metric. (It should be possible to read this chapter after Chapter 1, with the intervening chapters as reference.) Finding an associated torsion free and metric compatible bimodule connection (or quantum Levi-Civita connection, QLC for short) on the cotangent bundle is a non-linear problem which we solve directly in a variety of examples. Even the 2 x 2 matrices provide a nontrivial moduli of curved geometries. The chapter also covers Connes’ spectral triples sometimes in a weakened form but with the ‘Dirac operator’ now geometrically realized by a connection and a Clifford structure. Other topics include a wave-operator approach to quantum Riemannian geometry that bypasses the quantum metric and connection themselves. We end with a slightly different theory of Hermitian-metric compatible connections and Chern connections in noncommutative geometry.