Viewpoint Of Differential Geometry Research Articles

On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems

In the present paper, we discuss Monge-Ampère equations from the viewpoint of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space and that its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we show that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equations.

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.

GENERAL SCHWARZ LEMMAS BETWEEN PSEUDO-HERMITIAN MANIFOLDS AND HERMITIAN MANIFOLDS

<abstract> From the viewpoint of differential geometry, Schwarz lemmas of distance-decreasing type and volume-decreasing type can be obtained by the estimates of sum functions and product functions of all eigenvalues of holomorphic maps. This paper investigates general Schwarz lemmas by estimating partial sum functions and partial product functions of the eigenvalues of generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. </abstract>

Target Detection Through Riemannian Geometric Approach With Application to Drone Detection

Radar detection of small drones in presence of noise and clutter is considered from a differential geometry viewpoint. The drone detection problem is challenging due to low radar cross section (RCS) of drones, especially in cluttered environments and when drones fly low and slow in urban areas. This paper proposes two detection techniques, the Riemannian-Brauer matrix (RBM) and the angle-based hybrid-Brauer (ABHB), to improve the probability of drone detection under small sample size and low signal-to-clutter ratio (SCR). These techniques are based on the regularized Burg algorithm (RBA), the Brauer disc (BD) theorem, and the Riemannian mean and distance. Both techniques exploit the RBA to obtain a Toeplitz Hermitian positive definite (THPD) covariance matrix from each snapshot and apply the BD theorem to cluster the clutter-plus-noise THPD covariance matrices. The proposed Riemannian-Brauer matrix technique is based on the Riemannian distance between the Riemannian mean of clutter-plus-noise cluster and potential targets. The proposed angle-based hybrid-Brauer technique uses the Euclidean tangent space and the Riemannian geodesical distances between the Riemannian mean, the Riemannian median and the potential target point. The angle at the potential target on the manifold is computed using the law of cosines on the manifold. The proposed detection techniques show advantage over the fast Fourier transform, the Riemannian distance-based matrix and the Kullback-Leibler (KLB) divergence detectors. The validity of both proposed techniques are demonstrated with real data.

Information and contact geometric description of expectation variables exactly derived from master equations

In this paper a class of dynamical systems describing expectation variables exactly derived from continuous-time master equations is introduced and studied from the viewpoint of differential geometry, where such master equations consist of a set of appropriately chosen Markov kernels. To geometrize such dynamical systems for expectation variables, information geometry is used for expressing equilibrium states, and contact geometry is used for nonequilibrium states. Here time-developments of the expectation variables are identified with contact Hamiltonian vector fields on a contact manifold. Also, it is shown that the convergence rate of this dynamical system is exponential. Duality emphasized in information geometry is also addressed throughout.

Intrinsic curvature and topology of shadows in Kerr spacetime

From the viewpoint of differential geometry and topology, we investigate the characterization of the shadows in a Kerr spacetime. Two new quantities, the length of the shadow boundary and the local curvature radius are introduced. Each shadow can be uniquely determined by these two quantities. For the black hole case, the result shows that we can constrain the black hole spin and the angular coordinate of the observer only by measuring the maximum and minimum of the curvature radius. While for the naked singularity case, we adopt the length parameter and the maximum of the curvature radius. This technique is completely independent of the coordinate system and the location of the shadow, and is expected to uniquely determine the parameters of the spacetime. Moreover, we propose a topological covariant quantity to measure and distinguish different topological structures of the shadows.

Gliding Robotic Fish: An Underwater Sensing Platform and Its Spiral-Based Tracking in 3D Space

AbstractGliding robotic fish are a new type of underwater robot that combines the advantages of energy efficiency of underwater gliders and high maneuverability of robotic fish. Tail-enabled spiraling, as a novel locomotion pattern of gliding robotic fish, uses a buoyancy-driven mechanism and features a small turning radius. This paper investigates the spiral trajectory characteristics from the viewpoint of differential geometry and exploits them for curve tracking in the 3D space. The influences of control inputs on spiral trajectories are investigated through both simulation and experiments. A simulation example using a combined feedforward and feedback controller illustrates the proposed curve-tracking approach.

Finsler geometry of topological singularities for multi‐valued fields: Applications to continuum theory of defects

Topological singularity in a continuum theory of defects and a quantum field theory is studied from a viewpoint of differential geometry. The integrability conditions of singularity (Clairaut‐Schwarz‐Young theorem) are expressed by a torsion tensor and a curvature tensor when a Finslerian intrinsic parallelism holds for the multi‐valued function. In the context of the quantum field theory, the singularity called an extended object is expressed by the torsion when the intrinsic parallelism is related to the spontaneous breakdown of symmetry. In the continuum theory of defects, the path‐dependency of point and line defects within a crystal is interpreted by the non‐vanishing condition of torsion tensor in a non‐Riemannian space osculated from the Finsler space, and the domain is not simply connected. On the other hand, for the rotational singularity, an energy integral (J‐integral) around a disclination field is path‐independent when a nonlinear connection is single‐valued. This means that the topological expression for the sole defect (Gauss‐Bonnet theorem with genus ) is understood by the integrability of nonlinear connection.image

Finding a cyclide given three contact conditions

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belong to a codimension one subset, then there is a one-parameter family of solutions.

Geometric interpretation of linear viscoelasticity and characterization of chain architecture of polymers

We suggest a new interpretation on time-temperature superposition from the viewpoint of differential geometry and invented a viscoelastic function, called intrinsic phase angle, which is the arc length of the curve in plot of sinδ versus cosδ. The intrinsic phase angle gives a new master curve which is equivalent to conventional one without determination of shift factor. We also developed new plots based on the intrinsic phase angle and applied them to characterization of chain architecture of nonlinear polymers. Although new plots of intrinsic phase angle are similar to van Gurp-Palmen plot, they show the relation between structures of complex fluids and their rheology more clearly.

Geometries for possible kinematics

The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are revealed. Almost all geometries fall into pairs. There exists $t \leftrightarrow 1/(\nu^2t)$ correspondence in each pair. In the viewpoint of differential geometry, there are only 9 geometries, which have right signature and geometrical spatial isotropy. They are 3 relativistic geometries, 3 absolute-time geometries, and 3 absolute-space geometries.

Differential geometric approach to the stress aspect of a fault: Airy stress function surface and curvatures

We considered the two-dimensional stress aspect of a fault from the viewpoint of differential geometry. For this analysis, we concentrated on the curvatures of the Airy stress function surface. We found the following: (i) Because the principal stresses are the principal curvatures of the stress function surface, the first and the second invariant quantities in the elasticity correspond to invariant quantities in differential geometry; specifically, the mean and Gaussian curvatures, respectively; (ii) Coulomb’s failure criterion shows that the coefficient of friction is the physical expression of the geometric energy of the stress function surface; (iii) The differential geometric expression of the Goursat formula shows that the fault (dislocation) type (strike-slip or dip-slip) corresponds to the stress function surface type (elliptic or hyperbolic). Finally, we discuss the need to use non-biharmonic stress tensor theory to describe the stress aspect of multi-faults or an earthquake source zone.

A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in ℝ 3 with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary value problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solution of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

The $tt*$ Structure of the Quantum Cohomology of CP1 from the Viewpoint of Differential Geometry

The $tt*$ Structure of the Quantum Cohomology of CP1 from the Viewpoint of Differential Geometry

On a lower and upper bound for the curvature of ellipses with more than two foci

Let a set of points in the Euclidean plane be given. We are going to investigate the levels of the function measuring the sum of distances from the elements of the pointset which are called foci. Levels with only one focus are circles. In case of two different points as foci they are ellipses in the usual sense. If the set of the foci consists of more than two points then we have the so-called polyellipses. In this paper we investigate them from the viewpoint of differential geometry. We give a lower and upper bound for the curvature involving explicit constants. They depend on the number of the foci, the rate of the level and the global minimum of the function measuring the sum of the distances. The minimizer will be characterized by a theorem due to E. Weiszfeld together with a new proof. Explicit examples will also be given. As an application we present a new proof for a theorem due to P. Erdős and I. Vincze. The result states that the approximation of a regular triangle by circumscribed polyellipses has an absolute error in the sense that there is no way to exceed it even if the number of the foci are arbitrary large.

Geodesics for Dual Connections and Means on Symmetric Cones

Means on self-dual and homogeneous cones (, i.e., symmetric cones) are discussed from a viewpoint of differential geometry with affine connections. We first define means on symmetric cones in an axiomatic way following [8]. Next we consider dualistic differential geometry (, i.e., Riemannian metric and affine connections) [1] naturally introduced on symmetric cones. Elucidating the relation between the geodesics defined by each affine connection, and operator monotone functions that generate means, we show an important class of means are expressed by the (mid)points on geodesics. Related results on the means and submanifolds in a symmetric cone are also presented.

Thermodynamic fundamental equation of contact lines: selection of independent variables

In this work we review the development of the generalized theory of capillarity. When considering the theory of contact lines, we find that the theory developed by Boruvka and Neumann (BN) requires significant modifications. Their choice of parameters (independent variables) for a line is insufficient for formulating the fundamental equation. Furthermore, there are differential geometric constraints on these geometric parameters but not included in their analysis. As a result, their independent variables are in fact dependent. To describe the geometry of contact lines properly, we present a new set of parameters from the differential geometry viewpoint and subsequently give the fundamental equation for the thermodynamic system of contact lines.

Improving Path Trimming in a Network Algorithm for Fisher's Exact Test in Two-way Contingency Tables

This article provides an improvement of the network algorithm for calculating the exact p value of the generalized Fisher's exact test in two-way contingency tables. We give a new exact upper bound and an approximate upper bound for the maximization problems encountered in the network algorithm. The approximate bound has some very desirable computational properties and the meaning is elucidated from a viewpoint of differential geometry. Our proposed procedure performs well regardless of the pattern of marginal totals of data.

About Hölder condition numbers and the stratification diagram for defective eigenvalues

In this paper, we look at a particular case of application which is Hölder continuous, namely the map from a matrix A to one of its eigenvalues λ, when it is multiple defective. Two asymptotic Hölder condition numbers are considered: one (with respect to the other) is associated with a generalization of the Fréchet (with respect to Gateaux) derivative [A. Harrabi, About Hölder condition numbers of Gateaux and Fréchet type for general nonlinear functions, Manuscript — CERFACS, 1998]. We illustrate on a fully developed 3×3 example why these asymptotic condition numbers may not be appropriate for analyzing eigencomputations performed in finite precision. We present the complementary view point of differential geometry, developed by A. Ilahi in [A. Ilahi, Validation du calcul sur ordinateur: application de la théorie des singularités, Ph.D. Thesis, Université des Sciences Sociales, Toulouse I, 1998], which is based on the stratification diagram in C 3×3 . The distance to the stratum indicates when this complementary viewpoint is required.

Molecular geometry and symmetry from a differential geometry viewpoint

Relations between an earlier generalization of molecular symmetry called symmorphy and a molecular equivalence based on diffeomorphisms of electron density functional graphs (the so-called DFG equivalence introduced in our previous work) are analyzed. Any two DFG-equivalent electron density functions can be derived from one another by a suitable transformation of the spatial coordinates and the electronic charge density scale; the classes of DFG equivalence are the orbits of a group of linear operators operating in the space of electron density functions. Within the symmorphy framework, the symmetry group is derived from the symmorphy group by taking an intersection of a subgroup of the symmorphy group and the group of isometries for a natural choice of the Riemannian metric tensor. The Riemannian metric properties provide a choice for a suitable reference electron density function for each class of equivalent densities. Such reference densities serve as tools for a systematic classification of the infinite family of electron densities of molecular conformations. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64: 669–678, 1997