We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.
It was known that if the Gaussian curvature function along each meridian on a surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is decreasing, then the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $θ^{-1}(0)$ is empty or a subarc of the opposite meridian $θ^{-1}(π).$ For example, the surface of revolution $(R^2, dr^2+m_0(r)^2dθ^2),$ where $m…
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to certain well-know surface of revolution.
For a specific class of surfaces of revolution S, the existence of a smooth map Φ from a neighbourhood U of S to the Euclidean plane E2 preserving distances infinitesimally along the meridians and the parallels of S and sending the meridional arcs of U \ S to straight lines of E2; is proven.
This note derives parametrizations for surfaces of revolution that satisfy an affine-linear relation between their respective curvature radii. Alongside, parametrizations for the uniform normal offsets of those surfaces are obtained. Those parametrizations are found explicitly for a countably-infinite many of them, and of those, it is shown which are algebraic. Lastly, for those surfaces which have a constant ratio of principal curvatures, parametrizations with a constant angle between the parameter curves are found.
In this paper we consider three dimensional upper half space $\mathbb{H}^3 $ equipped with various Kropina metrics obtained by deformation of hyperbolic metric of $\mathbb{H}^3$ through $1$-forms and obtain a partial differential equation that characterizes minimal surfaces immersed in it. We prove that such minimal surfaces can only be obtained when the hyperbolic metric is deformed along $x^3$ direction. Then we classify such minimal surfaces and show that flag curvature of these surfaces is always non-positive. We also obtain the geodesics of this surface. In particular, it follows that suc…
In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Besides, we introduce the finite Chen type surfaces of revolution with nonvanishing Gauss curvature with respect to the third fundamental form. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.
In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed $H$, the strategy consists in rewriting …
There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids, and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family {M_n}_n of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or ma…
详细分析了广义Weierstrass表示的旋转曲面及其变形引起的修改Korteweg-德弗里斯（mKdV）方程。特别是，它表明，这些变形保持环面。讨论了曲面势的几何意义，并描述了几何变形（Moebius反演和向对偶曲面的过渡）引起的曲面势的变形。引入了mKdV族第一积分作为曲面的不变量对周围空间度量共形变化的不变性猜想。一些数值计算也证实了这一点。本文是前一作者的论文“Modified Novikov-Veselov equation and differential geometry of surfaces”（dg-ga/9511005）的续篇，其中考虑了一般曲面。在这里，旋转曲面被视为一般情况下的玩具模型。
The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces
Geodesics on the Torus and other Surfaces of Revolution Clarified Using Undergraduate Physics Tricks with Bonus: Nonrelativistic and Relativistic Kepler Problems
