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Using the mathematical MAPLE package, we construct models of surfaces of revolution of constant Gaussian curvature and geodesic lines on such surfaces.

We give an elementary treatment of the curvature of surfaces of revolution in the language of vector calculus, using differentials rather than an explicit parameterization. We illustrate some basic...

We classify all surfaces with constant Gaussian curvature K in Euclidean 3-space that can be expressed by an implicit equation of type $$f(x)+g(y)+h(z)=0$$ , where f, g and h are real functions of one variable. If $$K=0$$ , we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If $$K\not =0$$ , we prove that the surface is a surface of revolution.

We describe surfaces of revolution with the parallel equal zones property, where equally-spaced planes parallel to the axis of rotation cut the surface into zones with equal surface areas. The sphe...

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 the equation for $H$, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for surfaces of revolution with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for $S$, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of $H$ and $S$., Comment: 25 pages (23 in the published version), 4 figures. Key-words: Lorentz-Minkowski space, surface of revolution, skew curvature, mean curvature, Lorentz number

This paper is concerned with establishing Lp estimates for a class of maximal operators associated to surfaces of revolution with kernels in Lq(Sn−1 × Sm−1), q > 1. These estimates are used in extrapolation to obtain the Lp boundedness of the maximal operators and the related singular integral operators when their kernels are in the L(logL)κ(Sn−1 × Sm−1) or in the block space $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$(Sn−1 × Sm−1). Our results substantially improve and extend some known results.

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or $$\mathbb {R}^N$$ .

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface, the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo&rsquo, s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R&times, S1 and describe in detail the geodesics behaviour, the conjugate and cut loci.

In the present work, the notion of generalized Cheng&ndash, Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng&ndash, Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng&ndash, Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semi-global Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions., Comment: 24 pages, 9 figures

In this work, the generalized parametric Marcinkiewicz integral operators with mixed homogeneity related to surfaces of revolution are studied. Under some weak conditions on the kernels, the boundedness of such operators from Triebel&ndash, Lizorkin spaces to L p spaces are established. Our results, with the help of an extrapolation argument, improve and extend some previous known results.

The centroid theorems of Pappus (or the Pappus–Guldin theorems, or the Guldin theorems) show deep connections between areas, arc lengths, volumes of revolution, surfaces of revolution, and centers ...

We establish the bounds of Marcinkiewicz integrals associated to surfaces of revolution generated by two polynomial mappings on Triebel–Lizorkin spaces and Besov spaces when their integral kernels are given by functions $\unicode[STIX]{x1D6FA}\in H^{1}(\text{S}^{n-1})\cup L(\log ^{+}L)^{1/2}(\text{S}^{n-1})$. Our main results represent improvements as well as natural extensions of many previously known results.

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space H² into surfaces of revolution N that was initiated in [12, 13]. When the target N = H² we proved in [12] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps., National Science Foundation (U.S.) (Grant DMS-1302782), National Science Foundation (U.S.) (Grant 1045119)

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere., Comment: 7 pages

In this work, we study the $$L^p$$ estimates for a certain class of rough maximal functions with mixed homogeneity associated with the surfaces of revolution. Using these estimates with an extrapolation argument, we obtain some new results that represent substantially improvements and extensions of many previously known results on maximal operators.

The paper develops a study of closed geodesics on piecewise smooth constant curvature surfaces of revolution initiated by I.V. Sypchenko and D. S. Timonina. The case of constant negative curvature is considered. Closed geodesics on a surface formed by a union of two Beltrami surfaces are studied. All closed geodesics without self-intersections are found and tested for stability in a certain finite-dimensional class of perturbations. Conjugate points are found partly.

Let $\mathcal{S}$ be a surface of revolution embedded in the Heisenberg group $\mathcal{H}$. A revolution ring $R_{a,b}(\mathcal{S})$, $0, Comment: 15 pages, correct statement and proof of the main theorem

A generalized constant ratio surface (GCR surface) is definedby the property that the tangential component of the position vector is aprincipal direction at each point on the surface, see [8] for details. In thispaper, by solving some differential equations, a complete classificationof Lorentz GCR surfaces in the three-dimensional Minkowski space ispresented. Moreover, it turns out that a flat Lorentz GCR surface is anopen part of a cylinder, apart from a plane and a CMC Lorentz GCRsurface is a surface of revolution. 1. IntroductionThe concept of constant slope surfaces is introduced by Munteanu in [15],which are the surfaces whose normal makes a constant angle with the positionvector. In particular, Munteanu gave a nice characterization of constant slopesurfaces in Euclidean 3-space. Motivated by Munteanu’s work, constant slopesurfaces in Minkowski 3-space R 31 were classified by the authors in [10, 11].On the other hand, B. Y. Chen introduced in [3] the concept of constant ratiosubmanifolds, which is defined by the property that the ratio of the length ofthe tangential and normal components of its position vector is constant. Chen[3] also obtained the classification of constant ratio hypersurfaces in R

For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones., Comment: 24 pages, 9 figures

We solve the forward and inverse problems associated with the transformation of flat sheets with circularly symmetric director fields to surfaces of revolution with non-trivial topography, including Gaussian curvature, without a stretch elastic cost. We deal with systems slender enough to have a small bend energy cost. Shape change is induced by light or heat causing contraction along a non-uniform director field in the plane of an initially flat nematic sheet. The forward problem is, given a director distribution, what shape is induced? Along the way, we determine the Gaussian curvature and the evolution with induced mechanical deformation of the director field and of material curves in the surface (proto-radii) that will become radii in the final surface. The inverse problem is, given a target shape, what director field does one need to specify? Analytic examples of director fields are fully calculated that will, for specific deformations, yield catenoids and paraboloids of revolution. The general prescription is given in terms of an integral equation and yields a method that is generally applicable to surfaces of revolution.

Consider in hyperbolic space ℍ3 the one parameter family of immersed (non embedded) constant mean curvature surfaces of revolution Dπ with constant mean curvature H > 1. The parameter π ∈ (−∞, 0) is the analogue of the “necksize” of the Delaunay surfaces in Euclidean space. It is proved that when π → -∞, there exists a branch of surfaces with constant mean curvature H which bifurcate from Dπ. Furthermore, we prove that these new surfaces have only a discrete group of symmetries. The proof consists in a detailed study of the behaviour of the eigenvalues of the Jacobi operator when π tends to − ∞, together the bifurcation theorem of Crandall-Rabinowitz.

We study the Gauss map G of surfaces of revolution in the 3-dimensional Euclidean space $${{\mathbb {E}}^3}$$ with respect to the so-called Cheng–Yau operator $$\square $$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map G satisfying $$\square G=AG$$ for some $$3\times 3$$ matrix A are the planes, right circular cones, circular cylinders, and spheres.

We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution. Розраховано енергію одиничного нормального векторного поля на рiмановiй поверхні M. Показано, що енергія одиничного нормального векторного поля не залежить від вибору ортогонального базиса в дотичному просторі. Визначено енергію поверхні. Більш того, розраховано енергію сфер, областей на прямому круговому циліндрі та торі і, більш загально, поверхонь обертання.

The problem of realizability as a surface of revolution embedded into ℝ3 is studied and solved for a two-dimensional Bertrand’s Riemannian manifold being a configuration space of an inverse problem of dynamics. The problem of local realizability (near a parallel) of those manifolds is also solved.

We study resonances of surfaces of revolution obtained by removing a disk from a cone and attaching a hyperbolic cusp in its place. These surfaces include ones with non-trapping geodesic flow (every maximally extended non-reflected geodesic is unbounded) and yet infinitely many long-living resonances (resonances with uniformly bounded imaginary part, i.e., decay rate).

In this paper, we establish the boundedness of rough singular integrals associated to surfaces of revolution generated by two polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces.

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric., Comment: 17 pages, 1 figure

A loxodrome is a curve that makes a constant angle with the meridians. We use conformal maps and the notion of parallel transport in differential geometry to investigate loxodromes on hypersurfaces of revolution and their spiral behavior near a pole.

Well-posed boundary-value problems in multiply-connected regions are targeted for some sets of two-dimensional Laplace equations written in geographical coordinates on joint surfaces of revolution. Those are problems that simulate potential fields induced by point sources in joint perforated thin shell structures consist of fragments of different geometry. A semi-analytical approach is proposed to accurately compute solutions of such problems. The approach is based on the matrix of Green's type formalism. The elements of required matrices of Green's type are obtained analytically and expressed in closed computer-friendly form. This makes it possible to efficiently deal with the targeted class of problems. Copyright © 2014 John Wiley & Sons, Ltd.

Magnetic fields in conducting liquids (in particular, magnetic fields of galaxies, stars, and planets) are described by the magnetic induction operator. In this paper, we study the spectrum and eigenfunctions of this operator on a compact two-dimensional surface of revolution. For large magnetic Reynolds numbers, the asymptotics of the spectrum is studied; equations defining the eigenvalues (quantization conditions) are obtained; and examples of spectral graphs near which these points are located are given. The spatial structure of the eigenfunctions is studied.

In the present paper, we study surfaces of revolution in the three dimensional pseudo-Galilean space G31 . Also, we characterize surfaces of revolution in G31 in terms of the position vector field and Gauss map.

In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space , satisfying the condition , where is the Laplace operator with respect to the second fundamental form, is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in which satisfy the condition , coincide with surfaces of revolution with non-zero constant Gaussian curvature.

We investigate two settings of the Ginzburg—Landau system posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg—Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg—Landau heat flow posed on certain surfaces of revolution with boundary.

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in I_{p}^{3} has to be timelike or lightlike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in I_{p}^{3}. As applications, we describe the surfaces of revolution which are the orbits of a plane curve under a hyperbolic rotation with constant Gaussian and mean curvature., 11 pages, 4 figures. All comments are welcome

We consider a maximization problem for eigenvalues of the Laplace–Beltrami operator on surfaces of revolution in R 3 with two prescribed boundary components. For every j , we show there is a surface Σ j that maximizes the j th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$ G_\mu(\alpha,\beta)=C_1(\mu(M)) \int_M \frac{\alpha}{\mu}\frac{\beta}{\mu}\,\mu + C_2(\mu(M)) \int_M\alpha \cdot \int_M\beta $$ for some smooth functions $C_1,C_2$ of the total volume $\mu(M)$. Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness., Comment: 13 pages, 3 figures. Adapted to the final accepted version

In this paper the authors study the boundedness for parabolic Littlewood-Paley operator , where and satisfies a condition introduced by Grafakos and Stefanov in [6]. The result in the paper extends some known results.

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.

Freely rising air bubbles in water sometimes assume the shape of a spherical cap, a shape also known as the big bubble. Is it possible to find some objective function involving a combination of a bubble's attributes for which the big bubble is the optimal shape? Following the basic idea of the definite integral, we define a bubble's surface as the limit surface of a stack of n frusta (sections of cones) each of equal thickness. Should the objective function's variables correspond to the n base lengths of the frusta, then the critical points of the objective function might yield an optimally shaped bubble for which the limit as n → ∞ exists. One simple objective function which appears to model the big bubble is a linear combination of the bubble's upper and lower surface areas. Furthermore, with a computer algebra system, we can see in real time the shape of these critical bubbles as we vary the parameters of the objective function. Such a modeling project is suitable for a vector calculus or num...

Three well-known classes of surfaces of revolution are considered. The problem of their intersection and existence of common parts is studied.

A power function is characterized by a certain constant volume ratio associated with the surface of revolution generated by the graph of the function. We generalize this characterization to include...

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.

A spherical ring is the object that remains when a cylindrical drill bit bores through a solid sphere along an axis, removing from the sphere a capsule consisting of a cylinder with a spherical cap on each end, as shown in Figure 1. Remarkably, the volume of such a spherical ring depends only on its height, defined as the height of its cylindrical inner boundary, and not on the radius of the sphere from which it was cut.

In this article, we study surfaces of revolution without par- abolic points in a Euclidean 3-space whose Gauss map G satisfies the condition ¢ h G = AG,A 2 Mat(3,R), where ¢ h denotes the Laplace op- erator of the second fundamental form h of the surface and Mat(3,R) the set of 3◊3-real matrices, and also obtain the complete classification the- orem for those. In particular, we have a characterization of an ordinary sphere in terms of it.

We study special infinitesimal geodesic deformations of the surfaces of revolution in the Euclidean space E 3.