  We consider a generic curved non-commutative torus extending the notion of
conformally deformed non-commutative torus from \cite{Connes-Tretkoff}. In
general, a curved non-commutative torus is no longer represented by a spectral
triple, not even by a twisted spectral triple. Therefore, the geometry of this
manifold is governed by a positive second order differential operator
(Laplace-Betrami operator) rather than a first order differential operator
(Dirac operator). For this manifold, we prove an asymptotic expansion of the
heat semi-group generated by Laplace-Beltrami operator and provide an algorithm
to compute the local invariants which appear as coefficients in the expansion.
This allows to extend the results of \cite{Connes-Tretkoff},
\cite{Connes-Moscovici}, \cite{FaKh} (beyond conformal case and/or for
multi-dimensional tori).
