Local invariants of noncommutative tori

The notion of a generic curved noncommutative torus is considered, which extends the notion of conformally deformed noncommutative torus introduced by Connes and Tretkoff. For this manifold, an asymptotic expansion is established for the heat semigroup generated by Laplace–Beltrami operator (in fact, for an arbitrary selfadjoint positive elliptic differential operator of order $2$) and an algorithm is provided to compute the local invariants that arize as coefficients in the expansion. This allows one to extend a series of previous results by several authors beyond the conformal case and/or for multidimensional tori.