Narrow Results

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil₃. Also, we construct helicoidal surfaces in Nil₃ with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

We show that, in the sense of Baire categories, a typical Alexandrov surface with curvature bounded below by κ has no conical points. We use this result to prove that, on such a surface (unless it is flat), at a typical point, the lower and the upper Gaussian curvatures are equal to κ and ∞, respectively.

A gauge theory of contact is presented, based on the general idea that the local deformation of the nucleon surface at contact should be gauged by the variation of curvature. A contact force is then defined so as to cope with both the variation of curvature and the deformation. This force generalizes the classical definition of surface tension, in that it depends on the mean curvature, but also depends on the variance of the second fundamental form of surface, considered as a statistical variable over the ensemble of contact spots. It turns out that the variance of the second fundamental form does not depend but on the metric of the space of curvature parameters, organized as Riemann space. This result compels us to review the definition of physical surface of a nucleon.

According to the theory of wavelet analysis on $R^2$, the wavelet analysis on a smooth surface $M$ with nonzero constant Gaussian curvature will be discussed systematically in this paper. First, a general area-preserving projection from a smooth surface to the plane will be presented by the Gaussian projection and the area-preserving projection on the sphere. Then the continuous wavelet transform and its inverse transform on a smooth surface $M$ with nonzero constant Gaussian curvature will be discussed by a general area-preserving projection, relative dilation operator and translation operator. Further, according to the multi-resolution analysis on a smooth surface, the discrete wavelet transform and relative properties will be investigated systematically, including the two-scale equations of the wavelet function, orthogonality and so on. Finally, two numerical examples will be given.

The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Bäcklund transformation are obtained.

The main result of this paper gives a characterization of left-invariant almost $\alpha$-coKähler structures on three-dimensional (3D) semidirect product Lie groups ${ℝ}^2{⋊}_Aℝ$ in terms of the matrix $A$. Then, we study the harmonicity of the Reeb vector field $\xi$ of a simply connected homogeneous almost $\alpha$-coKähler three-manifold, in terms of the Gaussian curvature of the canonical foliation.

The goal of this paper is to study SdS space-time and its gravitational field with consideration of the canonical form and invariants of the curvature tensor. The characteristic of $\lambda$-tensor identifies the type of gravitational field. Gaussian curvature quantities enunciated in terms of curvature invariants.

In this paper, we study the $n$-body problem in the Poincaré upper half-plane ${ℍ}_R^2$, where the radius $R$ of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on ${ℍ}_R^2$. We analyze the relative equilibrium of positive masses moving along geodesics under the $\text{SL}(2,ℝ)$ group. This result is utilized to establish the existence of relative equilibria for the $n$-body problem on ${ℍ}_R^2$ for $n=2$ and $n=3$. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature.

In this paper, we study the problem of constructing a family of surfaces (surface pencils) from a given curve in $4$-dimensional Euclidean space ${𝔼}^4$. We have shown that the generalized rotation surfaces in ${𝔼}^4$ are the special type of surface pencils. Further, the curvature properties of these surfaces are investigated. Finally, we give some examples of flat surface pencils in ${𝔼}^4$.

In this paper, we discuss the recent existence results and compactness of solutions in the study of Nirenberg's problem in the 90's. We will discuss some new results and will outline some key ideas used in proofs of Theorems old and new. We also present some new questions.