We classify all rotational surfaces in Euclidean space whose principal curvatures $\kappa_1$ and $\kappa_2$ satisfy the linear relation $\kappa_1=a\kappa_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces.

中文翻译：

---

满足主曲率线性关系的欧氏空间旋转曲面的分类

我们对欧几里得空间中主曲率 $\kappa_1$ 和 $\kappa_2$ 满足线性关系 $\kappa_1=a\kappa_2+b$ 的所有旋转曲面进行分类，其中 $a$ 和 $b$ 是两个常数。我们根据其生成曲线给出了这些曲面的变分表征。作为我们分类的结果，我们发现具有类似于 Delaunay 曲面的几何行为的封闭（嵌入和非嵌入）曲面和周期性（嵌入和非嵌入）曲面。