  We compute the Ricci curvature of a curved noncommutative three torus. The
computation is done both for conformal and non-conformal perturbations of the
flat metric. To perturb the flat metric, the standard volume form on the
noncommutative three torus is perturbed and the corresponding perturbed
Laplacian is analyzed. Using Connes' pseudodifferential calculus for the
noncommutative tori, we explicitly compute the second term of the short time
heat kernel expansion for the perturbed Laplacians on functions and on 1-forms.
The Ricci curvature is defined by localizing heat traces suitably.
Equivalerntly, it can be defined through special values of localized spectral
zeta functions. We also compute the scalar curvatures and compare our results
with previous calculations in the conformal case. Finally we compute the
classical limit of our formulas and show that they coincide with classical
formulas in the commutative case.
