A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex structure, and the curvature is explicitly calculated. A noncommutative analogue of the fact that the catenoid is a minimal surface is studied by constructing a Laplace operator from the connection and showing that the embedding coordinates are harmonic. Furthermore, an integral is defined and the total curvature is computed. Finally, classes of left and right modules are introduced together with constant curvature connections, and bimodule compatibility conditions are discussed in detail.

中文翻译：

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非交换链状体

与经典链状体相对应的非交换代数与微分推导一起被引入。我们证明存在与复杂结构兼容的唯一度量和无扭连接，并明确计算曲率。通过从连接构造拉普拉斯算子并显示嵌入坐标是调和的，研究了悬链线是最小表面这一事实的非交换类比。此外，定义了一个积分并计算了总曲率。最后，介绍了左右模块的类别以及恒定曲率连接，并详细讨论了双模块兼容性条件。