标题： 曲率、面积和莫尔球的高斯-博内公式 显示英文标题

标题： Curvature, area and Gauss-Bonnet formula of the Moyal sphere

摘要： 我们研究了Moyal球的一些几何性质。 使用普通空间中球的共形度量和矩阵基，我们计算了Moyal球的标量曲率、总曲率积分和面积。 我们发现当非对易参数趋近于0时，Moyal球的标量曲率和面积会恢复为普通球的值。 随着非对易参数的增加，Moyal球的面积将减少并最终趋近于0。 我们发现二维Moyal球的总曲率积分仍然满足通常的高斯-博内公式，并且不依赖于非对易参数。 我们还计算了具有常曲率的共形度量的近似表达式，并得到了相应的修正函数。 此外，我们研究了一种具有两个非对易参数的广义变形Moyal球，并得到了类似的结果。 显示英文摘要

摘要： We studied some geometric properties of the Moyal sphere. Using the conformal metric of the sphere in ordinary space and the matrix basis, we calculated the scalar curvature, total curvature integral and area of the Moyal sphere. We found that when the noncommutative parameter approaches to 0, the scalar curvature and area of the Moyal sphere return to those of the ordinary sphere. As the noncommutative parameter increases, the area of the Moyal sphere will decrease and eventually approach to 0. We found that the total curvature integral of the two-dimensional Moyal sphere still satisfies the usual Gauss-Bonnet formula and does not depend on the noncommutative parameter. We also calculated the approximate expression of the conformal metric with a constant curvature and obtained the corresponding correction function. In addition, we studied a type of generalized deformed Moyal sphere with two noncommutative parameters and obtained similar results.