Given an arbitrary \(C^\infty \) Riemannian manifold \(M^n\), we consider the problem of introducing and constructing minimal hypersurfaces in \(M\times \mathbb {R}\) which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space \(\mathbb {R}^3=\mathbb {R} ^2\times \mathbb {R}\). Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in \(M\times \mathbb {R}\) have the same fundamental uniqueness properties of the helicoids in \(\mathbb {R}^3.\) We provide several examples of properly embedded vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on \(\mathrm{Nil}_3\) and \(\mathrm{Sol}_3\) are also presented. We show that vertical helicoids of \(M\times \mathbb {R} \) whose horizontal sections are totally geodesic in M are locally given by a “twisting” of a fixed totally geodesic hypersurface of M. We give a local characterization of hypersurfaces of \(M\times \mathbb {R}\) which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in \(M\times \mathbb {R}\) if and only if M admits families of isoparametric hypersurfaces. If so, properly embedded vertical catenoids can be constructed through the solutions of a certain first-order linear differential equation. Finally, we give a complete classification of the hypersurfaces of \(M\times \mathbb {R}\) whose angle function is constant.

给定任意\（C ^ \ infty \）黎曼流形\（M ^ n \），我们考虑在具有相同基本属性的\（M \ times \ mathbb {R} \）中引入和构造最小超曲面的问题欧氏空间\（\ mathbb {R} ^ 3 = \ mathbb {R} ^ 2 \ times \ mathbb {R} \）的标准螺旋线和链线形。通过在其高度函数和水平截面上施加条件来定义此类超曲面，然后将其称为垂直螺旋面和垂直链状面。我们确定\（M \ times \ mathbb {R} \）中的垂直螺旋线具有与\（\ mathbb {R} ^ 3。\）中的螺旋线相同的基本唯一性。在M是简单连接的空间形式之一的情况下，我们提供了几个正确嵌入的垂直螺旋面的示例。还介绍了垂直螺旋线，它们是\（\ mathrm {Nil} _3 \）和\（\ mathrm {Sol} _3 \）上函数的完整图。我们表明的该垂直螺旋面\（M \倍\ mathbb {R} \） ，其水平部分是在全测中号通过的固定全测超曲面的“扭转”被局部给予中号。我们给出\（M \ times \ mathbb {R} \）的超曲面的局部特征，其高度函数的梯度作为主要方向。结果，我们证明了垂直链状体存在于\（M \ times \ mathbb {R} \）当且仅当M接受等参超曲面族。如果是这样，则可以通过某个一阶线性微分方程的解来构造正确嵌入的垂直悬链线。最后，我们给出角函数为常数的\（M \ times \ mathbb {R} \）的超曲面的完整分类。