In this paper, we study cylindrical helices, Mannheim curves and Darboux developable surfaces associated to Mannheim curves. We give a method for constructing Mannheim curves from spherical curves and demonstrate that all Mannheim curves can be constructed by such a way. We also introduce the spherical Darboux images of Mannheim curves, and we give a classification of singularities of Darboux developable surfaces and spherical Darboux images. Moreover, we study the geometric properties of singularities as applications of singularity theory for spherical curves.

中文翻译：

---

曼海姆曲线和球面曲线

在本文中，我们研究了与曼海姆曲线相关的圆柱螺旋、曼海姆曲线和 Darboux 可展曲面。我们给出了一种从球面曲线构造曼海姆曲线的方法，并证明了所有曼海姆曲线都可以用这种方式构造。我们还介绍了曼海姆曲线的球面Darboux图像，并对Darboux可展曲面和球面Darboux图像的奇点进行了分类。此外，我们将奇点的几何特性作为奇点理论在球面曲线中的应用进行研究。