We will start by a review of classical facts from spectral geometry that are important for defining local geometric invariants of noncommutative spaces. Following the Gauss-Bonnet theorem of Connes and Tretkoff for the noncommutative two torus (proved in 2009), remarkable attention has been paid in recent years to the calculation and study of geometric invariants appearing in the heat kernel expansions of Laplacians on noncommutative spaces. After providing an overview of these developments, main aspects of my joint work with Alain Connes will be presented, in which the noncommutave analog of the term with explicit information about the Riemann tensor is calculated and studied in detail.
