Curvature in Noncommutative Geometry

Understand

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years.

• This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus.
• Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator.
• To carry explicit computations however one needs quite intriguing new ideas.