【摘要】 Nil₃空间是带有Heisenberg群结构的黎曼空间.利用Heisenberg群的群算子构造Nil₃空间中的仿射平移曲面.仿射平移曲面是由两条平面曲线作为基线通过群运算生成的.由于群运算不具有交换性,选定一组基线可以生成两类不同的仿射平移曲面.之后对这两种仿射平移曲面进行分类.利用常数变易法和欧拉待定指数函数法解曲面的平均曲率等于零时所对应的微分方程，并给出不同运算下极小仿射平移曲面的分类定理.最后给出一些具体的极小仿射平移曲面，并用Mathematica画出相应的图像.更多还原

【Abstract】 Nil₃ space is a Riemannian space with a Heisenberg group structure. The affine translation surfaces in Nil₃ space are constructed by using the group operator of the Heisenberg group. These surfaces are generated by two planar curves as base lines through group operation.However, since group operations are not commutative, selecting the same base lines generates two types of different affine translation surfaces. Then, the two affine translation surfaces are classified. Employ the method of variation of constants and Euler’s method of indeterminate exponential functions to solve the differential equation arising when the mean curvature of a surface equal zero, and present the classification theorem for minimal affine translation surfaces under various operations. Finally, some specific minimal affine translation surfaces are given, and corresponding images are drawn using Mathematica.更多还原