Quantum Riemannian Structures

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.