Einstein-Hilbert作用与黎曼非对易环面的Gauss-Bonnet定理
Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori
摘要：
我们证明了共形平坦黎曼度量的Einstein-Hilbert作用的非正性。只有当度量是常数平坦时，动作才消失。这恢复了Fathizadeh-Khalkhali在非交换四环面上的谱三元组的设置中的早期结果。进而给出了基于模算子的梯度流和该空间的数量曲率的计算。我们还证明了高斯-博内定理的参数化类的非对角度量的非交换两个环面。
英文摘要：
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.
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