Differential Geometry Research Papers - Academia.edu (original) (raw)

A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we... more

- by Paul Watts
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- Mathematics, Lie Algebra, Physics, Quantum Mechanics

We give an explicit construction of any simply-connected superconformal surface phicolonM2toR4\phi\colon M^2\to \R^4phicolonM2toR4 in Euclidean space in terms of a pair of conjugate minimal surfaces g,hcolonM2toR4g,h\colon M^2\to\R^4g,hcolonM2toR4. That phi\phiphi is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs (g,h)(g,h)(g,h) of conjugate minimal surfaces that give rise to images of holomorphic curves

Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous connection to the symmetric Yano connection is obtained on a normal almost contact manifold with Norden metric and closed structural 1-form. The curvature properties of this connection are studied on two basic classes of normal almost contact manifolds with Norden metric.

- by Marta Teofilova
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- Differential Geometry
- by Eugenie Hunsicker
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- Mathematical Physics, String Theory, Differential Geometry, Pure Mathematics
- by Cláudio Silva
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- Computer Graphics, Animation, Computational Geometry, Image-based Rendering
- by Athanase Papadopoulos
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- Differential Geometry, Global Analysis, Geometry, Analysis on Manifolds
- by Alison Hale
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- Engineering, Mathematics, Physics, Electromagnetism
- by Necip Erdoğan
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- Differential Geometry

AbstractThe work extends the A. Connes’ noncommutative geometry to spaces withgeneric local anisotropy. We apply the E. Cartan’s anholonomic frame approachto geometry models and physical theories and develop the nonlinear connectionformalism for projective modulespaces. Examples of noncommutative generationof anholonomic Riemann, Finsler and Lagrange spaces are analyzed. We alsopresent a research on noncommutative Finsler–gauge theories, generalized Finslergravity and anholonomic (pseudo) Riemann geometry which appear naturally ifanholonomic frames (vierbeins) are deﬁned in the context of string/M–theoryand extra dimension Riemann gravity..Pacs: 02.40.Gh, 02.40.-k, 04.50.+hMSC numbers: 83D05, 46L87, 58B34, 58B20, 53B40, 53C07 Contents 1 Introduction 22 Commutative and Noncommutative Spaces 42.1 Algebras of functions and (non) commutative spaces . . . . . . . . . . . 52.2 Commutative spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Noncommutative spaces . . . . . ....

- by Sergiu Vacaru
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- Mathematical Physics, Quantum Cosmology, High Energy Physics, Differential Geometry

A geometric version of the Poincar\'e Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications include generalizations of the Intermediate Value Theorem and Rolle's Theorem.

- by Jenny Harrison
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- Differential Geometry

In this paper we study the topology of the space of harmonic maps from S 2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

- by Arleigh Crawford
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- Differential Geometry, Pure Mathematics
- by Andrei Agrachev
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- Differential Geometry, Pure Mathematics

The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor and immune cells. The model consists of differential equations with piecewise constant arguments and based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is obtained a system of difference equations from the system of differential equations. In order to get local and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a consequence of Neimark-Sacker bifurcation.

In this paper we study the second fundamental form of translation surfaces in E3. We give a non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII. We classify those translation surfaces for which KII and H are proportional. Finally we obtain that there are no II-minimal translation surfaces in the Euclidean 3-space.

- by Marian Ioan Munteanu
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- Differential Geometry, Pure Mathematics, Mean curvature, Gaussian Curvature
- by valerii dryuma
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- Mathematics, Differential Geometry, Differential equation, Dynamic System

This work dates from January 2016 as a result of my remarks related to physics made during my high studies. I try in this work to explain the cause behind the inability of Newtonian mechanics to describe correctly many phenomena where the studied object rotates at a very high linear speed. I proved that, in this case, the velocity field is not equiprojective and that the famous formula for changing the reference frame is not correct. I made an application to the case of the GPS system satellites, then I presented a new method for studying a rotating system velocity without needing the conventional steps of changing reference frames. I finished my work by demonstrating the formulas of the main differential operators and I presented them with all the related steps and calculations by using the elementary surfaces. I am eager to discuss the results of this work further with physics and mathematics specialists, and I hope that my formulas will help to simplify the study of many difficult physics phenomena.

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended their 'speculations' when he declared that time was the fourth dimension. Yet our commonly perceived space is truly four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five-dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.

The differential geometry studies properties on curves and surfaces using tools from calculus and linear algebra.
We have the classic differential geometry on curves and surfaces as set out by Gauss in his Discisiones circa superficies curves., and the modern differential geometry of manifolds, the natural generalization of curves and surfaces in higher dimensions, through the calculus of tensors.FROM MY BOOOK "the relativity of geometry and physical space Amazon www.mpantes.gr

- by George Mpantes
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- Differential Geometry
- by Franz-Erich Wolter
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- Cognition, Differential Geometry, Image Construction
- by Cenap Ozel
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- Differential Geometry

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.

- by Andre Diatta
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- Mathematics, Computer Science, Computer Vision, Image Analysis

5th International Conference on Applied Mathematics and Sciences (AMA 2021)will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications impacts and challenges of Mathematics and Sciences. The conference documents practical and theoretical results which make a fundamental contribution for the development of Mathematics and Sciences. The aim of the conference is to provide a platform to the researchers and practitioners from both academia as well as industry to meet and share cutting-edge development in the field. The goal of this Conference is to bring together researchers and practitioners from academia and industry to focus on Mathematics and Sciences advancements, and establishing new collaborations in these areas. Original research papers, state-of-the-art reviews are invited for publication in all areas of Mathematics.

The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U(n), U(2n)/Sp(n)U(2n)/Sp(n)U(2n)/Sp(n) and U(n)/O(n)U(n)/O(n)U(n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of Sp(2)Sp(2)Sp(2) by categorical open sets. The obstacles to generalize these results to Sp(n)Sp(n)Sp(n) are discussed.

- by antonio Gómez-Tato
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- Differential Geometry, Pure Mathematics, Linear Model, Eigenvalues

It becomes clear that a mathematician persuaded of the truth of non-Euclidean geometry and seeking to convince others is almost driven to start by looking for, or creating, non-Euclidean three-dimensional Space, and to derive a rich theory of non-Euclidean two-dimensional Space from it — as Bolyai and Lobachevskii did, but not Gauss. The only hint that he explored the non-Euclidean three-dimensional case is the remark by Wachter, but what Wachter said was not encouraging: “Now the inconvenience arises that the parts of this surface are merely symmetrical, not, as in the plane, congruent; or, that the radius on one side is infinite and on the other imaginary” and more of the same. This is a long way from saying, what enthusiasts for Gauss’s grasp of non-Euclidean geometry suggest, that this is the Lobachevskian horosphere, a surface in non-Euclidean three-dimensional Space on which the induced geometry is Euclidean. In particular, there is no three-dimensional differential geometry l...

- by Jeremy Gray
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- Physics, Differential Geometry, Mathematics and its applications
- by Necip Erdoğan
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- Differential Geometry

Position Vectors of Space Curves has important applications in numerous mathematical field, thus we study it in an arbitrary space curves according to curvatures as constant with respect to Frenet frame in Euclidean 4-space 4 E

- by Fathi Math
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- Differential Geometry
- by Michael Gage
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- Mathematics, Differential Geometry, Pure Mathematics, Geodesic
- by Jean Ponce
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- Computer Vision, Differential Geometry, Scale Space, Structural Change

We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivation for this choice using light pressure. Mathematics Subject Classification (2000) : 83C05, 83C40, 83C99.

- by Maurice Dupre
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- Mathematics, Physics, General Relativity, Differential Geometry
- by Ratbay Myrzakulov
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- Differential Geometry

This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric... more

- by Mancho Manev
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- Differential Geometry, Pure Mathematics

We give a complete solution of a problem in submanifold theory posed and partially solved by the eminent algebraic geometer Pierre Samuel in 1947. Namely, to determine all pairs of immersions of a given manifold into Euclidean space that have the same Gauss map and induce conformal metrics on the manifold. The case of isometric induced metrics was solved in 1985 by the first author and D. Gromoll.

- by Ruy de Figueiredo
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- Differential Geometry, Euclidean space

En el siglo XVIII, el físico-matemático suizo L. Euler comenzó a desarrollar el método del Cálculo Variacional, que llegó a convertirse en uno de los instrumentos más importantes tanto en Matemáticas como en Física y que tendría después importantes repercusiones en otras áreas como la economía y la conocida optimización dinámica. El propósito de Euler se concentraba en encontrar aquellas curvas con longitudes ya sea máxima o mínima que cumplieran ciertas condiciones iniciales (como pueden ser los valores frontera fijos). De allí que tiempo después Joseph Louis Lagrange aplico varios de los métodos de Euler a ciertos problemas de optimización entre los cuales se encontraba uno muy particular el cual consistía en encontrar una superficie que realizase un mínimo del funcional área y que tuviese valores frontera fijos, que tiempo después sería llamado Problema de Plateau en honor al físico belga Joseph Plateau quien experimento con películas de jabón. Con todo ello, la historia de las superficies minimales tiene su inicio con L. Lagrange, el cual en su memoria \textit{Essai d’une nouvelle méthode pour déterminer les maxima et minima des formules intégrales indéfinies}, desarrollo un interesante algoritmo para el cálculo de variaciones que origino lo que hoy conocemos como la ecuación diferencial de Euler–Lagrange. De esta manera surge la teoría de superficies minimales en mathbbR3\mathbb{R}^3mathbbR3. Dicho problema fue planteado tiempo después como encontrar aquellas superficies que tuvieran una curvatura media nula llamadas superficies minimales. Aunque para Lagrange también era importante estudiar aquellas con curvatura media constante. En 1776 Jean Baptiste Marie Meusnier encontró que el plano, el helicoide y la catenoide eran soluciones al problema, además mostro que la ecuación de Euler-Lagrange podía ser modificada a una que se vería relacionada con la expresión de curvatura media. Tiempo después sería un caso particular de la ecuación diferencial de Monge-Ampère. Gaspard Monge y Legendre en 1795 dedujeron las fórmulas para representar las superficies solución. Schwarz encontró la solución del problema de Plateau para un cuadrilátero regular en 1865 y para un cuadrilátero general en 1867 (permitiendo la construcción de sus familias de superficies periódicas) utilizando métodos complejos. Weierstrass y Enneper desarrollaron fórmulas de representación más útiles, enlazando firmemente las superficies minimales al análisis complejo y a las funciones armónicas. La solución completa del problema de Plateau por Jesse Douglas y Tibor Radó fue un hito importante. El descubrimiento en 1982 por Celso Costa de una superficie que refutaba la conjetura de que el plano, la catenoide, y el helicoide son las únicas superficies minimales completas embebidas en mathbbR3\mathbb{R}^{3}mathbbR3 de tipo topológico finito. Actualmente la teoría de superficies minimales se ha diversificado a otros ambientes geométricos, adquiriendo importancia en la física matemática (por ejemplo en la conjetura de masa positiva, o en la conjetura de Penrose) y en la geometría de tres variedades (por ejemplo, en la conjetura de Smith, en la conjetura de Poincaré, y en la Conjetura de Geometrización de Thurston)

- by Chenxu He
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- Differential Geometry, Boolean Satisfiability
- by Zeki Kasap
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- Differential Geometry
- by Luc Florack
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- Computer Vision, Differential Geometry, Medical Image Analysis, Higher Order Thinking
- by Sergiu Vacaru
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- Mathematical Physics, Quantum Gravity, Differential Geometry, Linear Algebra

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where

- by Huai-Dong Cao
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- Algebraic Topology, Differential Geometry, Symplectic geometry, Degeneration
- by Harold Stalford
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- Applied Mathematics, Optimal Control, Differential Geometry, First-Order Logic