We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.

中文翻译：

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非交换环面上函数度量的谱几何


我们在非交换圆环上引入了一系列新的度量，称为函数度量，并研究了它们的谱几何。我们为这些指标定义了一类拉普拉斯型算子，并研究从热迹渐近中获得的它们的谱不变量。获得了热迹第二密度的公式。特别是，对于某些类别的度量（包括共形平坦度量和平坦度量的扭曲积），在所有维度上显式计算函数度量的标量曲率密度和总标量曲率。最后证明了配备通用函数度量的非交换两个环面的Gauss-Bonnet型定理。