Nassar Hassan Abdel all

• Doctor of Philosophy in mathematics
• Professor (Full) at Assiut University

The aim of this paper is to compute all the Frenet apparatus of non-transversal intersection curves (hyper-curves) of three implicit hypersurfaces in Euclidean 4-space. The tangential direction at a transversal intersection point can be computed easily, but at a non-transversal intersection point, it is difficult to calculate even the tangent vecto...

In this paper, motion of Darboux vector on two different space curves in Euclidean 3-space is investigated. The structure of the motion is based on ruled surfaces generated by Darboux vector w. According to this, developability of the considered ruled surfaces are studied. An extensive comparison between these developable ruled surfaces is performe...

In the present paper, the differential-geometrical framework for parallel bivariate Pareto distribution surfaces is given. Curvatures of a curve lying on , are interpreted in terms of the parameters of P. Geometrical and statistical interpretations of some results are introduced and plotted.

In this paper, we consider the problem of how to compute the Frenet apparatus of the non-transversal intersection curves (hyper-curves) of three hypersurfaces (given by their implicit-implicit-parametric and implicit-parametric-parametric equations) in Euclidean 4-space. The non-transversal intersection (in which the normal vectors of the intersect...

In this paper, a new representation of a cyclic surface by using circles of curvature of a space curve is presented. The conditions on the space curve such that cyclic surface with zero or non zero constant Gaussian curvature are given. The zero second Gaussian curvature is investigated. Also, minimal cyclic surfaces are studied. A procedure to det...

In this paper, a three dimensional surface using equiform motion of a surface of revolution in Euclidean 3-space E 3 is generated. The main results obtained in this paper are that the surface foliated by equiform motion of sphere has a zero scaler curvature if the motion of sphere are in parallel planes. Also, the surface foliated by equiform motio...

The purpose of this paper is to present algorithms for computing all the differential geometry properties of non-transversal intersection curves of three parametric hypersurfaces in Euclidean 4-space. For transversal intersections, the tangential direction at an intersection point can be computed by the extension of the vector product of the normal...

Riemann extension for the Anti Mach metric is derived, the solution of geodesic equations for the extended space are given and some properties for the extended space was studied and compared with the basic space.

Kinematics of moving generalized curves in a n-dimensional Euclidean space is formulated in terms of intrinsic geometries. The evolution equations of the orthonormal frame and higher curvatures are obtained. The integrability conditions for the evolutions are given. Finally, applications in R2 are given and plotted.

In this paper, the equations of motion for a general helix curve (τ = βκ) are derived by applying the first compatibility conditions for dependent variables (time and arc length). As application of the equations of motions, mkdv equation is solved using symmetry method.

Edges characterize boundaries are the problem of fundamental importance in description of geometric surfaces. Since edge detection is in the fore front of computer vision system for detection of surfaces images as Revolution and Saddle surfaces needs to description, it is crucial to have a good under standing of geometric properties of Surfaces and...

Kinematic of moving generalized space curves in Rn is formulated in terms of intrinsic geometries. The model for the dynamics is specified by accelera-tion fields. The acceleration is assumed to be local in the sense that it is a functional of the heir curvatures and their derivatives. By solving the nonlin-ear partial di®erential equations which g...

We present algorithms for computing the differential geometry properties of Frenet apparatus (t,n,b1,b2,κ1,κ2,κ3)(t,n,b1,b2,κ1,κ2,κ3) of intersection curves of implicit–parametric–parametric and implicit–implicit–parametric hypersurfaces in R4R4, for transversal intersection. Some examples are given and plotted.

The purpose of the present work is to construct new geometrical models for the motion of plane and space curves using an approach different from the one proposed by R. Mukherjee and R. Balakrishnan [1]. This approach is applied to a pair of coupled nonlinear partial differential equations (CNLPDEs) governing the time evolution of the curvature and...

We present algorithms for computing the di§erential geometry properties of Frenet apparatus of intersection curves of two implicit surfaces in R3 ; for transversal and tangential intersection using the implicit function theorem. We obtain a classiÖcation of the singularities on the intersection curve. Some examples are given and plotted.

The purpose of the present work is to construct a Hasimoto surface from its fundamental form coefficients via numerical integration of Gauss-Weingarten equations and fundamental theorem of surfaces.

This book is motivated by surface-surface intersection problems (SSI) in R, self-intersection surfaces problem in R and three hypersurfaces intersection problems (SSSI) in R . It presents algorithms for computing all the differential geometry properties of Frenet apparatus (t, n, b, ) of intersection curves of surfaces in R for transversal and tang...

We present algorithms for computing the differential geometry properties of Frenet apparatus {t, n, b, κ, τ} and higher-order derivatives of intersection curves of implicit and parametric surfaces in R 3 for transversal and tangential intersection. This work is considered as a continuation to Ye and Maekawa [1]. We obtain a classification of the si...

In this review, we try to answer the following question why should one study differential geometry? First of all, differential geometry is a Jewel of Mathematics. It is a prerequisite for theoretical physics. Secondly, in recent years, new and important applications have been discovered. Surprisingly, the structures of differential geometry are ide...

In this work, Jacobi elliptic function solutions for integrable nonlinear equations using F-expansion method are represented. KdV and Boussinesq equations are considered and new results are obtained.

The purpose of the present work is to construct new geometrical models for motion of plane curves. We have obtained nonlinear partial differential equations and have discussed the solutions of these equations using symmetry groups methods. Also, geometric interpretation for these solutions are given through the Gaussian and mean curvatures to the s...

The purpose of the present work is to construct new geometrical models for motion of plane curve by Darboux transformations. We get nonlinear partial differential equations (PDE). We have obtained the exact solutions of the resulting equations using symmetry groups method. Also, the Gaussian and mean curvatures of Monge form of the soliton surfaces...

In this paper, using the tanh-function method, we introduce a new approach to solitary wave solutions for solving nonlinear PDEs. The proposed method is based on adding integration constants to the resulting nonlinear ODEs from the nonlinear PDEs using the wave transformation. Also, we use a transformation related to those integration constants. So...

This paper mainly studies the singularities of Gauss Map of pedal hypersurface in Rn+1. It contains the geometry of pedal hypersurfaces in Rn+1 and their Gauss maps. The singularity of Gauss map of the pedal hypersurface using the rank of jacobian matrix of Gauss map is given and classified. The sets of singularities and its graphs under the Gauss...

This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss maps corresponding to the family of mappings are studied and the shape of these families and their singularities using mathematica program are illustrated and plotted. [M. A. Soliman, Nassar. H. Abdel-All, Soad. A. Hassan and E. Dahi. Families...

In this paper, we study the linear variations on a surface represents the configuration space of an equiform motion in R3. Therefore, the types of variations on a submanifold in Rn are given. Special types of variations are studied and plotted.

This article presents the geometry of middle surface of special line congruences, by calculating the metric quantities, determinants of first and second fundamental forms and mean curvature of the line congruences. These invariants are given in special cases for coordinate patches on reference surfaces of the incident line congruences.

This paper is concerned with the geometry of configuration space of a generalized type of similarity Helical motions (equiform kinematics). This study is considered as a continuation to [N. H. Abdel-All and F. Y. Mofarreh, Far East J. Math. Sci. (FJMS) 29, No. 2, 431–438 (2008; Zbl 1153.53006)]. For carrying out this investigation, we use the metho...

This article presents the intrinsic geometry of three neighbouring surfaces, by calculating the first fundamental quantities through the metric of these surfaces. The movement on these surfaces is described by frames given with respect to the frame of reference surface. These types of surfaces are plotted to confirm the theoretical results.

This paper is concerned to the geometry of configuration space of a generalized type of similarity motions (equiform kinematics). For carrying out this investigation, we use the tools of vector calculus to perform the computations. Finally, qualitative study on the singularities of the motion is given through some of the applications which are plot...

The aim of the present paper is to exhibit a connection between ridge or ravine on a hypersurface in Euclidean space ℝ n+1 , and the singularities of the caustic generated by the hypersurface normals. This work is a generalization to the results about a generic surface in ℝ 3 [E. V. Anoshkina, A. G. Belyaev and T. L. Kunii, Int. J. Shape Model. 1,...

In this paper, we present a differential geometric local study of two-parameter spatial motions: we look for points, which (up to the second order) instantaneously move on one-dimensional point paths, further we locally characterise these motions, which move ∞1 points in these way.

In this paper, we study cyclic surfaces in E 5 generated by equiform motions of a circle. The properties of this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E 5 in general are contained in canal hypersurfaces. Finally we give an example.

One of the most interesting and profound aspects of classical differential geometry is its interplay with the calculus of variations. In fact, the main differential geometric ideas of the calculus of variation occur over and over again and are continually being invented and rediscovered in a vast array of classical and modern differential geometry....

In this article, a new type of ruled surfaces in a Lorentz 3-space R13 is obtained by a strictly connected timelike oriented line moving with Frenet’s frame along a spacelike curve. These surfaces are classified into timelike and spacelike surfaces. The well-known theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved fo...

In this paper, we investigate motions of the 7-parameter group of equiform transformations with the property that three points move on three cir-cles with axes in one plane. We give an algorithm to find the corresponding one-parametric motion. It can be displayed as a curve in the space of motion parameters. As in general there seems to be no globa...

The differential-geometrical framework for analyzing statistical problems related to Pareto distribution, is given. A classical and intuitive way of description the relationship between the differential geometry and the statistics, is introduced [Publicationes Mathematicae Debrecen, Hungary, vol. 61 (2002) 1–14; RAAG Mem. 4 (1968) 373; Ann. Statist...

In the first part an overview on fuzzy sets and fuzzy numbers is given. A detailed treatment of these notions is introduced in [1,2,3]. This sintetically presentation is useful in understanding and in developping the applications in context problems. In the second part, fuzzy context model is given as an application of fuzzy sets and the fuzzy equi...

A representation of a special type of the variational problem on a surface immersed in a hyperbolic space is given. The perturbations of the mean curvature functional are studied. For this study, the corresponding frames are constructed and the polar surfaces of a given surface is established. The technique adapted here is based on Cartan’s methods...

Information geometry (Geometry and Nature) has emerged from the study of invariant properties of the manifold of probability distributions. It is regarded as a mathematical discipline having rapidly developing areas of applications as well as giving new trends in geometrical and topological methods. Information geometry has many applications belong...

We give a representation of the variational problem on a time (space) like surface immersed in a hyperbolic space. The geometric properties of the deformed surfaces are given. The variational problem for the Klein images of 2-parametric continuous motion of a line (kinematic surface) are introduced. The theory of Klein images is applied to a time l...

A brief account of information geometry and the deep relationship between the differential geometry and the statistics is given [N.H. Abdel-All, International Conference on Differential Geometry and its Applications, Cairo University, 19–26 June, Egypt, 2001; Springer Lecture Notes in Statistics, vol. 28, 1985; Math. Syst. Theory 20 (1987) 53]. The...

As it is well known that the most useful method of studying the properties of a curve in a Euclidean space from the standpoint of differential geometry is to make use of the Frenet formulas, in which the curvatures are the essential quantities for the curve. So, the motivation of the present work is to develop the variational problem in our work [C...

Some geometers have been interested in differential geometry of the variational problems connected with general surfaces. During the last few decades, this interest increased rapidly as more researchers became involved and gained results. Specifically one may cite, in this regard, the works of B.Y. Chen [J. London Math. Soc. 6 (2) (1973) 321; Total...

It is shown that the Petrov type-D for perfect fluid space-time with stiff matter can be obtained from S. Haggag and F. Desokey [Classical Quantum Gravity 13, No. 12, 3221–3228 (1996; Zbl 0875.83039)] and N. Dadhich [Gen. Relativ. Gravitation 28, No. 12, 1455–1463 (1996; Zbl 0865.53071)] by taking the limit when pressure tends to mass density. Some...

As is well known, the most useful method of studying the properties of a curve in a Euclidean space, from the standpoint of differential geometry, is making use of the Frenet formulas, in which the curvatures are the essential quantities for the curve. So, the motivation of the present work is to develop the variational problem in our work1 by usin...

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence CN. The mixed middle curvature and mixed curvature on CN are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study...

A surface M isometrically immersed in a Euclidean m-space Em is said to be of null 2-type if its immersion is obtained by harmonic functions and eigenfunctions of the Laplacian ∆ associated with a nonzero eigenvalue. Chen proved that surfaces in E 3 are of null 2-type if and only if they are open portions of circular cylinders.

This article presents a new distribution of Clifford Klein manifolds. Special kinds of the distribution, under some assumption, are studies. The geometrical properties of the K-manifolds in the considered distribution are given. The relations between Gauss, mean, scalar normal and Lipschitz-Killing curvatures are obtained. The methods adapted here...

This work is concerned mainly with the variational problem on an immersion x : M -+ E4. A new approach is in-troduced depends on the normal variation in any arbitrary normal direction in the normal bundle. The results of this work are consid-ered as a continuation and an extension to that obtained in [1]' [2] and [3], [4] respectively. The methods...

A surface M isometrically immersed in a Euclidean m-space Em is said to be of null 2-type if its immersion is obtained by harmonic functions and eigenfunctions of the Laplacian ∆ associated with a nonzero eigenvalue. Chen proved that surfaces in E 3 are of null 2-type if and only if they are open portions of circular cylinders. A surface is said to...

The aim of the present paper is an effort to make more exact some aspects of seven-parameter group of collineations in a five-dimensional Klein projective Space. Using the δ-variation of invariants in the first order contact elements, we derive several types of Klein manifolds on the Klein absolutum. Our study is carried out using Cartan's methods...

This work is devoted to the investigation of an m-submanifold, in the Euclidean n-space E n , foliated by a Euclidean (m-1)-plane. We extend results of N. Kuruoğlu and S. Keleş [Karadeniz Univ. Math. J. 6, 41–54 (1983; Zbl 0537.53006)] and of C. Thas [Yokohama Math. J. 26, 157–167 (1978; Zbl 0423.53045)]. The scalar normal curvature is obtained in...

This article is devoted to a description of some local differential geometric properties of immersions in a Klein hyperbolic space of dimension five. Using Klein representation, we extend the results in [7], some results of Lipschitz–Killing curvature, mean curvature vector and scalar normal curvature are given. The methods adapted here are based o...

The present work is concerned mainly with the study of two kinds of stratifiable pairs of line complexes embedded in an elliptic spaceS 3 as a continuation of [1] and [6], For both kinds, examples and some geometrical properties are given. Our study is carried out using Cartan's method of moving frames ([3], [8]).

THREE-DIMENSIONAL LINE MANIFOLDS WITH A SYMMETRIC MATRIX

CLASSIFICATION OF LINE COMPLEXES IN THE FLAG SPACE F 3

CLASS OF LINE COMPLEXES IN THE SECOND DIFFERENTIAL NEIGHBOURHOOD OF THE RAY IN THE FLAG SPACE F 3

A CLASSIFICATION OF LINE COMPLEXES IMMERSED IN QUASI-HYPERBOLIC SPACE “S 3 ’ THAT BASED ON THE MOBILITY OF A REPERE