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.

中文翻译：

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非交换三环的 Ricci 曲率

我们计算弯曲的非对易三环面的 Ricci 曲率。计算是针对平面度量的共形和非共形扰动完成的。为了扰动平面度量，对非交换三环面上的标准体积形式进行扰动，并分析相应的扰动拉普拉斯算子。对非交换环面使用 Connes 的伪微分演算，我们明确地计算了函数和 1 型上扰动拉普拉斯算子的短时热核展开的第二项。Ricci 曲率是通过适当地定位热迹来定义的。等效地，它可以通过局部谱 zeta 函数的特殊值来定义。我们还计算标量曲率，并将我们的结果与之前在共形情况下的计算结果进行比较。