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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 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 present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

Under consideration is the problem of reconstruction of a surface of revolution from the boundary curves of its projection. Two approaches to this problem are suggested. The first approach reduces the problem to a system of functional-differential equations. We describe in detail how to obtain this system. The second approach bases on geometrical considerations and uses a piecewiseconic approximation of the desired surface. The second method rests on the auxiliary statement on the 3D reconstruction of a straight circular cone. We give a formula for calculating the base radius of the cone. In the general case, the surface of revolution is approximated by the surface of rotation of some polygonal curve.

In this paper, we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution. We employ symbolic manipulations to derive the contact constraint equations from an ordered process, and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations (DAEs), leaving nine coupled differential equations, six contact equations, two holonomic constraint equations and four nonholonomic constraint equations. We then present a complete description of hands-free circular motions, in which the time-dependent variables are eliminated through a rotation transformation. We find that the circular motions, similar to those of the bicycle moving on a horizontal surface, nominally fall into four solution families, characterized by four curves varying with the angular speed of the front wheel. Then, we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface. Furthermore, we directly linearize the nonlinear DAEs, from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates. The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position. We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families. Finally, based on the numerical simulation of the original nonlinear DAEs system, we show that the stable circular motion is not asymptotically stable.

Aims of research. A surface of revolution is generated by rotation of a plane curve z = f(x) about an axis Oz called the axis of rotation. This paper provides information on hyperboloids of revolution surfaces and their classification. Their geometric modeling, linear and materially nonlinear analysis are worked out. Methods. Hyperboloids of revolution middle surface is plotted using the software MathCAD. The linear and materially nonlinear numerical analyses of thin shells of the shape of an hyperboloid of revolution surfaces on stress-strain state is given in this paper, using the finite elements method in a computer software R-FEM, the material which we use in our model is concrete with isotopic nonlinear 2D/3D stress-strain curve for materially nonlinear analysis and linear stress-strain curve for linear analyses. Comparison is done with the result of the finite elements linear analysis of their strain-stress results. Results. That displacements in the investigated shells subject to self-weight, wind load with materially nonlinear analysis are bigger than which done by linear analysis, in the other side the displacements is similarity subjected to free vibration load case. Based on these results, conclusions are made for the whole paper.

Semiclassical spectral series of the Laplace operator on a two-dimensional surface of revolution with a conical point are described. It is shown that in many cases asymptotic eigenvalues can be calculated from the quantization conditions on special Lagrangian tori, with the Maslov index of such tori being replaced by a real invariant expressed in terms of the cone apex angle.

This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface's gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields. Ministry of Education (MOE) This project was partially supported by Singapore Ministry of Education Tier 1 & Tier 2 Grants, National Natural Science Foundation of China Grants (No. 61702363, No. 61772312, No. 61802228), NSFC Joint Fund with Zhejiang Integration of Informatization and Industrialization under Key Project (U1609218) and the Key Research and Development Project of Shandong Province (2017GGX10110).

In this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible class is strictly below {4\pi}. Under a more restrictive but still explicit geometric smallness condition we obtain a quite interesting additional geometric information: The profile curve of this solution can be parameterised as a graph over the x-axis. By working below the energy threshold of {4\pi} and reformulating the problem in the Poincaré half plane, compactness of a minimising sequence is guaranteed, of which the limit is indeed smooth. The last step consists of two main ingredients: We analyse the Euler–Lagrange equation by an order reduction argument by Langer and Singer and modify, when necessary, our solution with the help of suitable parts of catenoids and circles.

This paper is devoted to the simulation of the mean curvature flow on a surface of revolution. The surface is discretized using the icosahedral decomposition, and the flow equation is discretized by the finite element method. The scheme stability is improved using spline interpolation.

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 $$\mathbb {R}^N$$ .

The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.

Calculus of variations and partial differential equations 60(1), 32 (2021). doi:10.1007/s00526-020-01875-6, Published by Springer, Heidelberg

In this paper, we classify conformal surfaces of revolution in hyperbolic3-space $\mathbb{H}^{3}(-c^{2})$ satisfying an equation in terms of the position vector field and the Laplace operators with respect to the first,the second and the third fundamental forms of the surface.

We extend the decoupling results of the first two authors to the case of real analytic surfaces of revolution in \({\mathbb R}^3\). New examples of interest include the torus and the perturbed cone.

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 article we consider a one-dimensional reaction-diffusion problem with mixed boundary conditions. We provide conditions for the existence or nonexistence of stable nonconstant solutions whose derivative vanishes at some point. As an application, we obtain similar results for problems with Dirichlet boundary conditions posed in some symmetric domains: an n-dimensional ball, surfaces of revolution, and model manifolds.

The paper studies surfaces with constant mean curvature (CMC) H. If H = 0 then the surfaces are minimal. CMC tori were studied by H. Wente. U. Abresz proved that Wente tori have one family of planar lines of curvature and characterized them with elliptic integrals.A.I. Bobenko in his studies considered the problem of constructing CMC tori E3, S3, H3. In this paper, CMC surfaces of revolution are investigated. For a surface in E3 the Bonnet’s theorem states that for any surface having constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature.According to this statement, for surfaces of revolution with constant positive Gaussian curvature, CMC surfaces are constructed using the Bonnet’s theorem. It is proved that constructed surfaces are also surfaces of revolution. A family of plane curvature lines (meridians) is described by elliptic integrals, and surfaces with Gaussian curvature are also described by elliptic integrals. These surfaces are constructed using the mathematical software package. DOI 10.14258/izvasu(2018)4-22

An invariant submodel of gas dynamics equations constructed on a three-dimensional subalgebra with a projective operator for the case of monatomic gas is under consideration. The submodel is reduced to an Abel equation, with integral curves constructed for it. For a separatrix of a saddle, an approximate solution is studied. Such solutions describe the vortex scattering of gas along plane curves placed on the surface of revolution.

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e. the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations, Ministerio de Economía y Competitividad

It is shown that the surface with a constant mean curvature encloses the extremal volume among all toroidal surfaces of given area. The exact solution for the corresponding variational problem is derived, and its parametric analysis is performed in the limits of high and small mean curvatures. An absence of smooth torus with constant mean curvature is proved, and the extremal surface is demonstrated to have at least one edge located on the outer side of the torus.

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)

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 article presents approaches to the formation of a general computational scheme for modeling (simulating) the particle motion on an axisymmetric rotating curved surface with a vertical axis of rotation. To describe the complex particle motion over a given surface, the fundamental equation of particle dynamics in a non-inertial reference frame was used, and by projecting it onto the axes of cylindrical coordinates, the Lagrange&rsquo, s differential equations of the first kind were obtained. According to the proposed algorithm in C#, an application was developed that enables graphical and numerical control of the calculation results. The program interface contains six screen forms with tabular baseline data (input) and a table of a step-by-step calculation of results (output), particle displacement, velocity, and acceleration diagrams constructed along the axes of the system of cylindrical coordinates &rho, and z, graphical presentation of the generate of the surface of revolution and the trajectory of the absolute motion of a particle over the axisymmetric rotating surface developed in polar coordinates. Examples of the calculation of the particle motion are presented. The obtained results can be used for the study and design of machines, for example, centrifugal rotary chopper machines.

This paper presents a new rigorous approach based on the method of analytical regularization developed for analysis of scalar wave diffraction by acoustically soft doubly-connected arbitrary shaped surfaces of revolution. The surface contains two apertures of equal angular size and is formed by rotation of a smooth generating curve. This approach is validated by comparison with previously known results.

In this article, we consider surfaces in the 3dimensional Euclidean space E3without parabolic points which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We present an important family of surfaces, namely, surfaces of revolution in E3. We study a special case of this family, namely, surfaces of revolution where the sum of the radii of the principal curvature R is constant.

In this paper, we investigate translating solitons for the inverse mean curvature flow (IMCF), which is a special solution deformed only for translation under the flow. The IMCF has been studied extensively not only as a type of a natural geometric flow, but also for obtaining various interesting geometric inequalities. We show that the translating solitons that are either ruled surfaces or translation surfaces are cycloid cylinders, and completely classify 2-dimensional helicoidal translating solitons and the higher dimensional rotationally symmetric translating solitons using the phase-plane analysis. The surface foliated by circles, which is called a cyclic surface, is regarded in terms of being the translating soliton for the IMCF, and then it is a surface of revolution whose revolution axis is parallel to the translating direction. In particular, we extend the result to a higher dimension, namely, the n-dimensional translating soliton foliated by spheres lying on parallel hyperplanes in $$\mathbb {R}^{n+1}$$ must be a rotationally symmetric hypersurface whose rotation axis is parallel to the translating direction.

Within the presented research rods exhibiting the maximum buckling resistance are looked for. It was assumed that they have to possess the same mass and the same length as a solid cylindrical bar treated as the reference one. Nonprismatic rods of circular hollow cross-section and of variable wall thickness were only included into considerations. It was assumed that the external and internal surfaces of revolution were defined by some assumed in advance smooth functions. Three different classes of functions were used in the research, and namely: polynomial of the second order, sine and hyperbolic cosine functions. The classical optimization problem was defined by the objective function which was the buckling force, being maximized with appropriate constraints. Design variables were three independent geometrical parameters defining uniquely the sought bar shape. All derivations were carried out analytically by means of MathematicaTM system. The presented procedure was illustrated by two examples. The obtained increase of the critical force was dependent on the slenderness of the reference bar and reached the value 40 in the case of the longer, considered reference bar.Within the presented research rods exhibiting the maximum buckling resistance are looked for. It was assumed that they have to possess the same mass and the same length as a solid cylindrical bar treated as the reference one. Nonprismatic rods of circular hollow cross-section and of variable wall thickness were only included into considerations. It was assumed that the external and internal surfaces of revolution were defined by some assumed in advance smooth functions. Three different classes of functions were used in the research, and namely: polynomial of the second order, sine and hyperbolic cosine functions. The classical optimization problem was defined by the objective function which was the buckling force, being maximized with appropriate constraints. Design variables were three independent geometrical parameters defining uniquely the sought bar shape. All derivations were carried out analytically by means of MathematicaTM system. The presented procedure was illustrated by two examples. The obtain...

International audience; The calculation of the diffusion-controlled reaction rate for partially absorbing, non-spherical boundaries presents a formidable problem of broad relevance. In this paper we take the reference case of a spherical boundary and work out a perturbative approach to get a simple analytical formula for the first-order correction to the diffusive flux onto a non-spherical partially absorbing surface of revolution. To assess the range of validity of this formula, we derive exact and approximate expressions for the reaction rate in the case of partially absorbing prolate and oblate spheroids. We also present numerical solutions by a finite-element method that extend the validity analysis beyond spheroidal shapes. Our perturbative solution provides a handy way to quantify the effect of non-sphericity on the rate of capture in the general case of partial surface reactivity.

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.

We study 2D Euler equations on a rotating surface, subject to the effect of the Coriolis force, with an emphasis on surfaces of revolution. We bring in conservation laws that yield long time estimates on solutions to the Euler equation, and examine ways in which the solutions behave like zonal fields, building on previous works that have examined how such 2D Euler equations can account for the observed band structure of rapidly rotating planets. Specific results include both an analysis of time averages of solutions and a study of stability of stationary zonal fields. The latter study includes both analytical and numerical work.

A topological classification, up to Liouville (leafwise) equivalence of integrable Hamiltonian systems given by flows with a smooth potential on two-dimensional surfaces of revolution is presented. It is shown that the restrictions of such systems to three-dimensional isoenergy surfaces can be modelled by the geodesic flows (without potential) of certain surfaces of revolution. It is also shown that in many important cases the systems under consideration are equivalent to other well-known mechanical systems. Bibliography: 29 titles.

We will show a symmetry result concerning Willmore surfaces of revolution which satisfy symmetric Dirichlet data. The first step is to find a regular energy minimizing surface. We will do this by carefully modifying a minimizing sequence using an idea by Dall’Acqua, Deckelnick & Grunau. After this we will establish a priori estimates for non-symmetric solutions, in which an order-reduction argument by Langer & Singer will be essential.

The paper presents a theoretical study of hypersingular equations of the general form for problems of electromagnetic-wave diffraction on open surfaces of revolution. Justification of the Galerkin is given. The method is based on the separation of the principal term and its analytic inversion. The inverse of the principal operator is completely continuous. On the basis of this result, the equivalence of the initial equation to a Fredholm integral equation of the second kind is proven. An example of numerical solution with the use of Chebyshev polynomials of the second kind is considered.

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

We consider the equation $\Delta u+f(u)=0$ on a surface of revolution with Dirichlet boundary conditions. We obtain conditions on $f$, the geometry of the surface and the maximum value of a positive solution in order to ensure its stability or instability. Applications are given for our main results.

This paper is denoted to further study of geodesic bifurcation on surfaces of revolution. We demonstrate an example of bifurcation of closed geodesics on surfaces.

The third chapter covers the first integral and the variational principle. The notions of cyclic coordinates and Poisson brackets are first recalled. The Theorem of Poisson, Euler equation, and the variational principle are then addressed. An application in optics, namely the Fermat principle, is reviewed. Five exercises are then solved, namely on the Watt Regulator, on the first integral of a free material point, on the brachistochrone problem, on the minimum surface of revolution, and on optical paths and Fermat principle.

Investigation of physics on two-dimensional curved surface has significant meaning in study of general relativity, inasmuch as its realizability in experimental analogy and verification of faint gravitational effects in laboratory. Several phenomena about dynamics of particles and electromagnetic waves have been explored on curved surfaces. Here we consider Wolf effect, a phenomenon of spectral shift due to the fluctuating nature of light fields, on an arbitrary surface of revolution (SOR). The general expression of the propagation of partially coherent beams propagating on arbitrary SOR is derived and the corresponding evolution of light spectrum is also obtained. We investigate the extra influence of surface topology on spectral shift by defining two quantities, effective propagation distance and effective transverse distance, and compare them with longitudinal and transverse proper lengths. Spectral shift is accelerated when the defined effective quantities are greater than real proper lengths, and vice versa. We also employ some typical SORs, cylindrical surfaces, conical surfaces, SORs generated by power function and periodic peanut-shell shapes, as examples to provide concrete analyses. This work generalizes the research of Wolf effect to arbitrary SORs, and provides a universal method for analyzing properties of propagation compared with that in flat space for any SOR whose topology is known., Comment: 23 pages, 7 figures

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed., 9 pages 11 figures

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.

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Mobius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.

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.

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.

This work aimed to study isocurvature surfaces of Earth’s normal gravity field and their relation to partial derivatives of a plumb line curvature. An isocurvature surface of a gravity field is a surface along which the value of the plumb line curvature is constant. The normal gravity field is a symmetrical gravity field; therefore, isocurvature surfaces are surfaces of revolution. To study an isocurvature surface, special assumptions are made to form a vector equation, which will hold only for a small coordinate patch of the isocurvature surface. The gradient of a normal plumb line curvature is vertical to the isocurvature surface pointing to the direction along which the curvature of the plumb line decreases or increases the most. In order to show the significance of isocurvature surfaces, it was shown that it is possible to determine the value of the surface derivative of a plumb line’s curvature without differentiating the original complicated function of a plumb line curvature. KeywordsPlumb Lines; Curvature; Normal Gravity Field; Plumbline Curvature; Isocurvature Surface

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.