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

This chapter continues the study of calculus on vector manifolds begun in Chapter 4. The emphasis here is on the central object of classical differential geometry, the curvature tensor. We have endeavored to supply simple and systematic derivations of all properties of the curvature tensor including relations to extrinsic geometry, behavior under transformations, and generalization to nonRiemannian curvature. We believe that some of our results are new, but our main objective is to demonstrate the unique advantages of the method and to develop the calculus to the point where application to any problem in differential geometry is straightforward.

An important problem in differential geometry and in gravitation is to identify metrics in a fully coordinate independent manner. In fact, the very foundation of Riemannian geometry is based on the existence of a tensor, the Riemann or curvature tensor, which vanishes if and only if the metric is locally flat. Many other such local characterizations of metrics are known. The aim of this article is to present a brief selection of them as an example of the fruitful interplay between differential geometry and gravity.

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.

The Weyl and Minkowski problems in differential geometry in the large

This does not attempt to be a systematic overview or to present a comprehensive list of problems. We outline some questions in three different areas which seem interesting to the author. Experts will learn little that is new; our goal is to give some picture of the fields for non-specialists.

A Problem in Differential Geometry

On the construction of minimal surfaces from geodesics

L'application des variétés à connexion à certains problèmes de la géométrie différentielle

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.

Kinetic or Boltzmann schemes are interesting alternatives to the macroscopic numerical methods for solving the hyperbolic conservation laws of gas dynamics. They utilize the particle-based description instead of the wave propagation models. While the continuous particle velocity based upwind schemes were developed in the earlier decades, the discrete velocity Boltzmann schemes introduced in the last decade are found to be simpler and are easier to handle. In this work, we introduce a novel way of introducing discrete velocities which correspond to the physical wave speeds and formulate a discrete velocity Boltzmann scheme for solving Euler equations.

Two-dimensional Riemann problems for conservation laws in gas dynamics

The differential geometry of circle systems has received a large amount of attention of late. The subject has been approached from two quite distinct sides. On the one hand we have Koenigs, Cosserat, Moore, Bompiani and others who fix their attention on what we may call the descriptive differential properties of circles. The totality of circles in three-dimensional space may be represented by an S6, Iying in an Ss, and this variety may be studied by the now familiar methods of projective diBerential geometry. The theorems so reached are invariant under the twenty-four parameter group of sphere transformations. The other class of writers, wherein we may include Bianchi, Tzitzeica, Guichard and Eisenhart, have confined themselves largely to congruences (two-parameter systems) of circles, frequently to normal congruences. The methods employed have been the general ones of differential geometry and the center of interest has been rather more in the axes of the circles than the circles themselves. It has seemed for some time to the present writer that the last word on these subjects had not by any means been written, and that by a different method of approach not only might we obtain simpler proofs of known theorems but discover a number of new theorems as well. The most interesting properties of circles are those which are invariant under inversion, or under the tenparameter group of conformal transformations of space; the best approach to this group is through the use of pentaspherical coordinates. It is true that some of the writers mentioned above have made use of these coordinates, and still more has been done by Darboux in his Theorie des Surfaces, yet the possibilities of these coordinates have been by no means exhausted, and in the present paper they are more systematically applied to problems in differential circle geometry than has been the case in the past. A circle may be regarded in two different aspects, either as a locus of points, or an envelop of spheres. The two points of view are, of course, closely related, but the change of emphasis leads naturally to rather different sets of theorems. The first section of the present article is devoted to preliminary formulse for points and spheres in pentaspherical coordinates, and certain

The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with a finite number of elements (polyhedra), the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory. The latter appears then as a limit of the refinement of the discretization. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization. One of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces, such as minimal, con stant mean curvature, isothermic surfaces, etc. Usually, one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties. The goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible. Our discretizations are based on quadrilateral meshes, i.e. we discretize parametrized surfaces. For the discretization of a special class of surfaces, it is natural to choose an adapted parametrization. In this paper, we investigate conformai discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figures 1 and 2. In comparison with direct methods (see, in particular, [23]), leading

Fluid dynamics research over the past twenty years has contributed immensely to our knowledge of atherosclerosis. The ability to detect localized atherosclerotic plaques using noninvasive ultrasonic methods was advanced significantly by investigations into the nature and occurrence of velocity disturbances created by arterial stenoses, and diagnosis of carotid bifurcation disease using a combination of ultrasonic imaging and Doppler measurement of blood velocity is now quite routine. Since atherosclerotic plaques tend to be localized at sites of branching and artery curvature and since these locations would be expected to harbor complex flow patterns, investigators postulated that fluid dynamics might play an initiating role in atherogenesis. Several fluid dynamic variables were proposed as initiating factors. Investigations were undertaken during the 1980s in which fluid dynamic model experiments with physiologic geometries and flow conditions were employed to simulate arterial flows and in which morphometric mapping of intimal thickness was performed in human arteries. Correlations between fluid dynamic variables and intimal thickness revealed that atherosclerotic plaques tended to occur at sites of low and oscillating wall shear stress; and these observations were reinforced by studies in a monkey model of atherosclerosis. Concomitantly, it was realized that arteries adapt their diameters so as to maintain wall shear stress in a narrow range of values around 15 dynes/cm2, findings which were based both on observations of normal arteries and on animal studies in which flow rates were manipulated and arterial diameter adaptation was measured.(ABSTRACT TRUNCATED AT 250 WORDS)

Parallel curves have been used extensively in a variety of fields during the last decade, including architecture, computer graphics, aerospace, and medicine. Despite these applications, the theoretical development of parallel curves in differential geometry has been relatively limited. In this study, we investigate parallel curves using the q-frame in three-dimensional Euclidean space, rather than the traditional Frenet frame. We introduce the new expression “quasi-parallel curves” as a generalization of classical parallel curves. We also examine several geometric properties of these curves and provide rigorous proofs. In differential geometry, ruled surfaces are of great importance. They are defined by the movement of generators, which generate straight lines on the surface. Furthermore, a directrix (base curve) is any curve that intersects all generators (rulings). In this paper, we investigate a new class of quasi-ruled surfaces whose base curve is both the original curve and its quasi-parallel curve in R 3 . We describe the geometric characteristics of these surfaces and derive the first and second fundamental forms, as well as the Gaussian and mean curvatures. Specific types of quasi-ruled surfaces are analyzed, and conditions for their developability and minimality are established. Several quasi-ruled surfaces generated from the helix curve as a base curve and others generated from its quasi-parallel curve as a base curve are provided, and the parametrization of the resulting surfaces are determined. Finally, all generated surfaces, together with their quasi-parallel and original curves, are visualized using Mathematica 13.