Problems In Differential Geometry Research Articles

Prescribed curvature measure problem in hyperbolic space

AbstractThe problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star‐shaped k‐convex bodies with prescribed (n‐k)‐th curvature measures by establishing crucial C2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.

On Disks Enclosed by Smooth Jordan Curves

Given a smooth-plane Jordan curve with bounded absolute curvature κ>0, we determine equivalence classes of distinctive disks of radius 1/κ included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely symmetric trajectory behaviour with respect to the curve turning. These lead to a decomposition of the plane into a finite number of maximal regions with respect to set inclusion leading to natural lower bounds for the length an area enclosed by the curve. We present a “half version” of the Pestov–Ionin theorem, and subsequently a generalisation of the classical Blaschke rolling disk theorem. An interesting consequence is that we describe geometric conditions relying exclusively on curvature and independent of any kind of convexity that allows us to give necessary and sufficient conditions for the existence of families of rolling disks for planar domains that are not necessarily convex. We expect this approach would lead to further generalisations as, for example, characterising volumetric objects in closed surfaces as first studied by Lagunov. Although this is a classical problem in differential geometry, recent developments in industrial manufacturing when cutting along some prescribed shapes on prescribed materials have revived the necessity of a deeper understanding on disks enclosed by sufficiently smooth Jordan curves.

Local and Global Properties of the Gravitational Lens Effect with Special Consideration of the Gravitational Lens Effect with Star Perturbation.

Since the late 1970s, gravitational lensing became an important tool in astrophysics, taking advantage of the lens-like bending of light by masses such as planets, stars, galaxies, or clusters of them to determine their properties or even their existence. At that time and later in the 80s, the group at the Hamburg observatory around Sjur Refsdal developed many techniques that are still in use to understand and apply the effect. Although the effect is a consequence of Einstein's general theory of relativity, the equations used to describe the effects of masses on light rays are relatively simple. However, in order to answer questions about what a light source looks like through a special lens, or whether there might be multiple images of a light source, the math got quite complicated and the problems were largely solved numerically.In this article we show, for an important special case of a star in a galaxy as a lens, that the problems of differential geometry that arise can be treated algebraically by a computer algebra system such as Maple and lead to elegant solutions that are generally applicable to mappings from the plane onto the plane.

Geometry and Geodesy on the Primary Visual Cortex as a Surface of Revolution

Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area V1, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical magnification factors. Once unfolded, V1 has a convex three-dimensional shape, which can be mathematically modeled as a surface of revolution embedded in three-dimensional Euclidean space. Thus, we solve the problem of differential geometry and geodesy for the mapping of the visual field to V1, involving both isotropic and non-isotropic cortical magnification factors of a most general form. We provide illustrations of our technique and results that apply to V1 surfaces with curve profiles relevant to vision research in general and to visual phenomena such as ‘crowding’ effects and eye movement guidance in particular. From a mathematical perspective, we also find intriguing and unexpected differential geometry properties of V1 surfaces, discovering that geodesic orbits have alternative prograde and retrograde characteristics, depending on the interplay between local curvature and global topology.

On Isoperimetric Problem in a 2-Dimensional Finsler Space of Funk type

The isoperimetric problem is one of the fundamental problems in differential geometry. By using the method of the calculus of variations we show that the circle centered at the origin in $${\mathbb {B}}^2(1)$$ is a proper maximum of the isoperimetric problem in a 2-dimensional Finsler space of Funk type. We also obtain the formula of area enclosed by a simple closed curve in a spherically symmetric Finsler plane.

Differential Invariants of One Parametrical Group of Transformations

The concept of differential invariant, along with the concept of invariant differentiation, is the key in modern geometry [1]-[10]. In the Erlangen program [3] Felix Klein proposed a unified approach to the description of various geometries. According to this program, one of the main problems of geometry is to construct invariants of geometric objects with respect to the action of the group defining this geometry. This approach is largely based on the ideas of Sophus Lee, who introduced continuous geometry groups of transformations, now known as Lie groups, into geometry. In particular, when considering classification problems and equivalence problems in differential geometry, differential invariants with respect to the action of Lie groups should be considered. In this case, the equivalence problem of geometric objects is reduced to finding a complete system of scalar differential invariants. The interpretation of the k- order differential invariant as a function on the space of k- jets of sections of the corresponding bundle made it possible to operate with them efficiently, and using invariant differentiation, new differential invariants can be obtained. Differential invariants with respect to a certain Lie group generate differential equations for which this group is a symmetry group. This allows one to apply the well-known integration methods to such equations, and, in particular, the Li- Bianchi theorem [4]. Depending on the type of geometry, the orders of the first nontrivial differential invariants can be different. For example, in the space R3 equipped with the Euclidean metric, the complete system of differential invariants of a curve is its curvature and torsion, which are second and third order invariants, respectively. Note that scalar differential invariants are the only type of invariants whose components do not change when changing coordinates. For this reason, scalar differential invariants are effectively used in solving equivalence problems. In this paper differential invariants of Lie group of one parametric transformations of the space of two independent and three dependent variables are studied. It is shown method of construction of invariant differential operator. Obtained results applied for finding differential invariants of surfaces.

A list of open problems in differential geometry

Good open problems play an indispensable role in the development of differential geometry. All speakers and participants of the conference Modern Trends in Differential Geometry (Sao Paulo, July 2018) were invited to include open problems. We thank the contributors and the organizers. We think that this collection reflects many facets of geometry, its deep roots, and its profusion of fruit.

A Transcendence Basis in the Differential Field of Invariants of Pseudo-Galilean Group

Let G be a subgroup in the group of all invertible linear transformations of a finite-dimensional real space X. One of the problems in differential geometry is that of finding easily verified necessary and sufficient conditions for G-equivalence of paths in X. In solving this problem, we use methods of the theory of differential invariants which give descriptions of transcendence bases of differential fields of G-invariant differential rational functions. Having explicit forms of transcendence bases, we can obtain efficient criteria for G-equivalence of paths with respect to actions of the special linear, orthogonal, pseudoorthogonal, and symplectic groups. We present a description of one finite transcendence basis in the differential field of differential rational functions invariant with respect to the action of the pseudo-Galilean group ΓO. Based on this, we establish necessary and sufficient conditions of ΓO-equivalence of paths.

Projectively Invariant Objects and the Index of the Group of Affine Transformations in the Group of Projective Transformations

The paper is grown from the lecture course "Metric projective geometry" which I hold at the summer school "Finsler geometry with applications" at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements, and also prove the following new statement: on a complete Riemannian manifold of nonconstant curvature the index of the group of affine transformations in the group of projective transformations is at most two.

Mapping superintegrable quantum mechanics to resonant spacetimes

We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

Euler, the clothoid and

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

On the construction of minimal surfaces from geodesics

In a recent article (2013), Li et al. [1] approximate minimal surfaces from geodesic boundaries, with applications to garment design in mind. We go over this work and existing methods for constructing minimal surfaces from geodesics. First, we justify why minimal surfaces and the problem of finding the surface with minimal area (i.e., solving Plateau's problem) have little to do with garment design. Second, we recall that Plateau's problem makes little sense for boundaries such as those considered in [1], composed of unclosed curves of finite length or disconnected pieces of them (with no other positional restriction). Finally, we note that the construction of a minimal surface (with zero mean curvature) from a prescribed geodesic is a particular instance of a classical problem in differential geometry, already solved by Björling. In particular, for a geodesic circle or helix the resulting minimal surfaces are well-known (catenoid and helicatenoid, respectively), so no approximations are required.

Selective orders in central simple algebras and isospectral families of arithmetic manifolds

Let k be a number field and B be a central simple algebra over k of dimension p 2 where p is prime. In the case that p = 2 we assume that B is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal orders of B with the property that a $${\mathcal{O}_{k}}$$ -order of rank p embeds into either every maximal order in the set or into none at all. Such a set is called nonselective. We prove upper and lower bounds for the cardinality of a maximal nonselective set. This problem is motivated by the inverse spectral problem in differential geometry. In particular we use our results to clarify a theorem of Vigneras on the construction of isospectral nonisometric hyperbolic surfaces and 3-manifolds from orders in quaternion algebras. We conclude by giving an example of isospectral nonisometric hyperbolic surfaces which arise from a quaternion algebra exhibiting selectivity.

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.

Proving isometry for homogeneous Einstein metrics on flag manifolds by symbolic computation

The question whether two Riemannian metrics on a certain manifold are isometric is a fundamental and also a challenging problem in differential geometry. In this paper we ask whether two non-Kähler homogeneous Einstein metrics on a certain flag manifold are isometric. We tackle this question by reformulating it into a related question on a parametric system of polynomial equations and answering it by carefully combining Gröbner bases and geometrical arguments. Using this technique, we are able to prove the isometry of such metrics.

A GEOMETRIC STUDY OF SOME EQUATIONS OF MATHEMATICAL PHYSICS

We introduce geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry, and find geometrical characteristics associated to equations of Mathematical Physics. Also, we introduce a geometric study of some boundary problems. Throughout this work, as main tool we employed an adequate Riemannian Hessian structure, suggested in [Int. J. Geom. Meth. Mod. Phys.7(7) (2010) 1104–1113].

ITERATIVE GEOMETRIC STRUCTURES

In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

Isometric Immersions and Compensated Compactness

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T

A fluid dynamic formulation of the isometric embedding problem in differential geometry

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.