Journal of Geometry and Symmetry in Physics Articles (Project Euclid)
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The latest articles from Journal of Geometry and Symmetry in Physics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Fri, 12 May 2017 12:16 EDTFri, 12 May 2017 12:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On Matrix Representations of Geometric (Clifford) Algebras
http://projecteuclid.org/euclid.jgsp/1494605778
<strong>Ramon G. Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 43, 1--36.</p><p><strong>Abstract:</strong><br/>
Representations of geometric (Clifford) algebras with real square matrices are reviewed by
providing the general theorem as well as examples of lowest dimensions. New definitions for
isometry and norm are proposed. Direct and indirect isometries are identified respectively
with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every
element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its
matrix representation of order $n$. It is deduced in which geometric algebras direct
isometries are inner automorphisms (similarity transformations of matrices). Indirect
isometries need reversion too. Finally, the most common isometries are reviewed in order to
write them in this way.
</p>projecteuclid.org/euclid.jgsp/1494605778_20170512121622Fri, 12 May 2017 12:16 EDTTwo Types of Lorentz Transformations for Massless Fieldshttp://projecteuclid.org/euclid.jgsp/1499306425<strong>Victor L. Mironov</strong>, <strong>Sergey V. Mironov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 83--96.</p><p><strong>Abstract:</strong><br/>
We describe a curious relationship between orthogonal spheres and causal
relationships in Minkowski space.
</p>projecteuclid.org/euclid.jgsp/1499306425_20170705220031Wed, 05 Jul 2017 22:00 EDTElastic Bending Energy: A Variational Approachhttps://projecteuclid.org/euclid.jgsp/1512442901<strong>Riccardo Capovilla</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 1--45.</p><p><strong>Abstract:</strong><br/>
Geometric continuum models for fluid lipid membranes are considered using classical field theory, within a covariant variational approach. The approach is cast as a higher-derivative Lagrangian formulation of continuum classical field theory, and it can be seen as a covariant version of the field theoretical variational approach that uses the height representation. This novel Lagrangian formulation is presented first for a generic reparametrization invariant geometric model, deriving its equilibrium equation, or shape equation, and its linear and angular stress tensors, using the classical Canham-Helfrich elastic bending energy for illustration. The robustness of the formulation is established by extending it to the presence of external forces, and to the case of heterogenous lipid membranes, breaking reparametrization invariance. In addition, a useful and compact general expression for the second variation of the free energy is obtained within the Lagrangian formulation, as a first step towards the study of the stability of membrane configurations. The simple structure of the expressions derived for the basic entities that appear in the mechanics of a lipid membrane is a direct consequence of the well established power of a Lagrangian variational approach. The paper is self-contained, and it is meant to provide, besides a new framework, also a convenient introduction to the mechanics of lipid membranes.
</p>projecteuclid.org/euclid.jgsp/1512442901_20171204220145Mon, 04 Dec 2017 22:01 ESTGeneralized Seiberg-Witten Equations on a Riemann Surfacehttps://projecteuclid.org/euclid.jgsp/1512442902<strong>Rukmini Dey</strong>, <strong>Varun Thakre</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 47--66.</p><p><strong>Abstract:</strong><br/>
In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten (S-W) equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra “Higgs field”. Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kähler manifold and construct a pre-quantum line bundle over the moduli space of solutions.
</p>projecteuclid.org/euclid.jgsp/1512442902_20171204220145Mon, 04 Dec 2017 22:01 ESTKähler Dynamics for the Universal Multi-Robot Fleethttps://projecteuclid.org/euclid.jgsp/1512442903<strong>Vladimir G. Ivancevic Ivancevic</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 67--85.</p><p><strong>Abstract:</strong><br/>
A general model is formulated for a universal fleet of all unmanned vehicles, including Aerial Vehicles (UAVs), Ground Vehicles (UGVs), Sea Vehicles (USVs) and Underwater Vehicles (UUVs), as a geometric Kähler dynamics and control system. Based on the Newton-Euler dynamics of each vehicle, a control system for the universal autonomous fleet is designed as a combined Lagrangian and Hamiltonian form. The associated continuous system representing a very large universal fleet is given in Appendix in the form of the Kähler-Ricci flow
</p>projecteuclid.org/euclid.jgsp/1512442903_20171204220145Mon, 04 Dec 2017 22:01 ESTGeodesics on Rotational Surfaces in Pseudo-Galilean Spacehttps://projecteuclid.org/euclid.jgsp/1512442904<strong>Dae Won Yoon</strong>, <strong>Murat Kemal Karacan</strong>, <strong>Bahaddin Bukcu</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 87--94.</p><p><strong>Abstract:</strong><br/>
In this paper, we study rotational surfaces in the pseudo-Galilean three-space $\mathbb G_3^1$ with pseudo-Euclidean rotations and isotropic rotations. In particular, we investigate properties of geodesics on rotational surfaces in $\mathbb G_3^1$ and give some examples.
</p>projecteuclid.org/euclid.jgsp/1512442904_20171204220145Mon, 04 Dec 2017 22:01 EST2+2 Moulton Configurationhttps://projecteuclid.org/euclid.jgsp/1512442905<strong>Naoko Yoshimi</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 95--110.</p><p><strong>Abstract:</strong><br/>
We pose a new problem of collinear central configurations in Newtonian $n$-body problem. It is known that the configuration of two bodies moving along the Newtonian force is always a collinear central configuration. Can we add new two bodies on the straight line of initial two bodies without changing the move of the initial two bodies and the configuration of the four bodies is central, too? We call it 2+2 Moulton configuration. We find three special solutions to this problem and find each mass of new two bodies is zero.
</p>projecteuclid.org/euclid.jgsp/1512442905_20171204220145Mon, 04 Dec 2017 22:01 ESTIs Spacetime as Physical as Is Space?https://projecteuclid.org/euclid.jgsp/1518577290<strong>Mayeul Arminjon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 1--24.</p><p><strong>Abstract:</strong><br/>
Two questions are investigated by looking successively at classical mechanics, special relativity, and relativistic gravity: first, how is space related with spacetime? The proposed answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. That space can be endowed with a natural structure of 3-D differentiable manifold, thus giving rise to a simple notion of spatial tensor -- namely, a tensor on the space manifold. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred-frame character? We find that it does not, for different physics (either relativistic or not) may be defined on the same spacetime structure - and also, the same physics can be implemented on different spacetime structures.
</p>projecteuclid.org/euclid.jgsp/1518577290_20180213220145Tue, 13 Feb 2018 22:01 ESTTwistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structureshttps://projecteuclid.org/euclid.jgsp/1518577291<strong>Ljudmila Kamenova</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 25--35.</p><p><strong>Abstract:</strong><br/>
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.
</p>projecteuclid.org/euclid.jgsp/1518577291_20180213220145Tue, 13 Feb 2018 22:01 ESTCommuting Pairs of Generalized Contact Metric Structureshttps://projecteuclid.org/euclid.jgsp/1518577292<strong>Janet Talvacchia</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 37--50.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the implications of this theorem with regard to the definitions of generalized Sasakian and generalized co-Kähler geometry.
</p>projecteuclid.org/euclid.jgsp/1518577292_20180213220145Tue, 13 Feb 2018 22:01 ESTA Note on the Class of Surfaces with Constant Skew Curvatureshttps://projecteuclid.org/euclid.jgsp/1518577293<strong>Magdalena Toda</strong>, <strong>Alexander Pigazzini</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 51--58.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to analyze surfaces with constant skew curvature (CSkC), and show that the class of CSkC surfaces with non-constant principal curvatures does not contain any Bonnet surfaces.
</p>projecteuclid.org/euclid.jgsp/1518577293_20180213220145Tue, 13 Feb 2018 22:01 ESTComposition Algebras, Exceptional Jordan Algebra and Related Groupshttps://projecteuclid.org/euclid.jgsp/1518577294<strong>Ivan Todorov</strong>, <strong>Svetla Drenska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 59--93.</p><p><strong>Abstract:</strong><br/>
Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.
</p>projecteuclid.org/euclid.jgsp/1518577294_20180213220145Tue, 13 Feb 2018 22:01 ESTFlat Affine and Symplectic Geometries on Lie Groupshttps://projecteuclid.org/euclid.jgsp/1518577295<strong>Andrés Villabón</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 95--121.</p><p><strong>Abstract:</strong><br/>
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe by Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.
</p>projecteuclid.org/euclid.jgsp/1518577295_20180213220145Tue, 13 Feb 2018 22:01 ESTIndefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruchhttps://projecteuclid.org/euclid.jgsp/1525939245<strong>Rolf-Peter Holzapfel</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 1--40.</p><p><strong>Abstract:</strong><br/>
In \cite{Ho02} we have counted indefinite metrics (two-dimensional, integrally defined, over Gauss numbers) with a fixed norm (discriminant). We would like to call them also \emph{indefinite class numbers}. In this article we change from Gauss to Eisenstein numbers. We have to work on the complex two-dimensional unit ball, an Eisenstein lattice on it and the quotient surface. It turns out that the compactified quotient is the complex plane $\IP^2$. In the first part we present a new proof of this fact. In the second part we construct explicitly a Heegner series with the help of Legendre-symbol coefficients. They can be interpreted as ``indefinite class numbers'' we look for. Geometrically they appear also as number of plane curves with (normed) Eisenstein disc uniformization.
</p>projecteuclid.org/euclid.jgsp/1525939245_20180510040050Thu, 10 May 2018 04:00 EDTCassini Ovals in Harmonic Motion Orbitshttps://projecteuclid.org/euclid.jgsp/1525939246<strong>Khristo N. Boyadzhiev</strong>, <strong>Irina A. Boyadzhiev</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 41--49.</p><p><strong>Abstract:</strong><br/>
We discover the appearance of interesting Cassinian ovals in the motion of a two-dimensional harmonic oscillator. The trajectories of the oscillating points are ellipses depending on a parameter. The locus of the foci of these ellipses is a Cassini oval. The form of this oval depends on the magnitude of the initial velocity.
</p>projecteuclid.org/euclid.jgsp/1525939246_20180510040050Thu, 10 May 2018 04:00 EDTNatural Coordinate System in Curved Space-Timehttps://projecteuclid.org/euclid.jgsp/1525939247<strong>Ying-Qiu Gu</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 51--62.</p><p><strong>Abstract:</strong><br/>
In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be presented according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it ``natural coordinate system''. The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In natural coordinate system, we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is helpful to understand the nature of space-time.
</p>projecteuclid.org/euclid.jgsp/1525939247_20180510040050Thu, 10 May 2018 04:00 EDTDeformations of Symplectic Structures by Moment Mapshttps://projecteuclid.org/euclid.jgsp/1525939248<strong>Tomoya Nakamura</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 63--84.</p><p><strong>Abstract:</strong><br/>
We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. We can deform a given symplectic structure $\omega $ with a Hamiltonian $G$-action to a new symplectic structure $\omega ^t$ parametrized by some element $t$ in $\Lambda^2\mathfrak{g}$. We can obtain concrete examples for the deformations of symplectic structures on the complex projective space and the complex Grassmannian. Moreover applying the deformation method to any symplectic toric manifold, we show that manifolds before and after deformations are isomorphic as a symplectic toric manifold.
</p>projecteuclid.org/euclid.jgsp/1525939248_20180510040050Thu, 10 May 2018 04:00 EDTSpanning Class in the Category of Braneshttps://projecteuclid.org/euclid.jgsp/1525939249<strong>Andrés Viña</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 85--104.</p><p><strong>Abstract:</strong><br/>
Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.
</p>projecteuclid.org/euclid.jgsp/1525939249_20180510040050Thu, 10 May 2018 04:00 EDTUnsteady Roto-Translational Viscous Flow: Analytical Solution to Navier-Stokes Equations in Cylindrical Geometryhttps://projecteuclid.org/euclid.jgsp/1527127300<strong>Alessio Bocci</strong>, <strong>Giovanni Mingari Scarpello</strong>, <strong>Daniele Ritelli</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 1--21.</p><p><strong>Abstract:</strong><br/>
We study the unsteady viscous flow of an incompressible, isothermal (Newtonian) fluid whose motion is induced by the sudden swirling of a cylindrical wall and is also starting with an axial velocity component. Basic physical assumptions are that the pressure axial gradient keeps its hydrostatic value and the radial velocity is zero. In such a way the Navier-Stokes PDEs become uncoupled and can be solved separately. Accordingly, we provide analytic solutions to the unsteady speed components, i.e., the axial $v_z(r,t)$ and the circumferential $v_\theta(r,t)$, by means of expansions of Fourier-Bessel type under time damping. We also find: the surfaces of dynamical equilibrium, the wall shear stress during time and the Stokes streamlines.
</p>projecteuclid.org/euclid.jgsp/1527127300_20180523220143Wed, 23 May 2018 22:01 EDTCentralizer of Reeb vector field in Contact Lie Groupshttps://projecteuclid.org/euclid.jgsp/1527127301<strong>Babak Hassanzadeh</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 23--31.</p><p><strong>Abstract:</strong><br/>
We consider the centralizer of Reeb vector field of a contact Lie group with a left invariant Riemannian metric while contact structure is left invariant. Then we decompose the Lie algebra of this Lie groups to centralizer of Reeb vector field and its orthogonal complement and using this decomposition the contact Lie group is investigated. Furthermore, in last section a special automorphism is defined and studied which it keeps the contact form.
</p>projecteuclid.org/euclid.jgsp/1527127301_20180523220143Wed, 23 May 2018 22:01 EDTAn Examination of Perpendicular Intersections of BFRS and MFRS in $E^3$https://projecteuclid.org/euclid.jgsp/1527127302<strong>Şeyda Kılıçoğlu</strong>, <strong>Süleyman Şenyurt</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 33--45.</p><p><strong>Abstract:</strong><br/>
We already have defined and found the parametric equations of Frenet ruled surfaces which are called Bertrandian Frenet Ruled Surfaces (BFRS) and Mannheim Frenet Ruled Surfaces (MFRS) of a curve $\alpha ,$ in terms of the Frenet apparatus. In this paper, we find a matrix which gives us all sixteen positions of normal vector fields of eight BFRS and MFRS in terms of the Frenet apparatus. Further using the orthogonality conditions of the eight normal vector fields, we give perpendicular intersection curves of the eight BFRS and MFRS.
</p>projecteuclid.org/euclid.jgsp/1527127302_20180523220143Wed, 23 May 2018 22:01 EDTSymmetry Groups of Systems of Entangled Particleshttps://projecteuclid.org/euclid.jgsp/1527127303<strong>Abraham A. Ungar</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 47--77.</p><p><strong>Abstract:</strong><br/>
A Lorentz transformation of signature $(m,n)$, $m,n\in\Nb$, is a pseudo-rotation in a pseudo-Euclidean space of signature $(m,n)$. Accordingly, the Lorentz transformation of signature $(1,3)$ is the common Lorentz transformation of special relativity theory. It is known that entangled particles involve Lorentz symmetry violation. Hence, the aim of this article is to expose and illustrate the symmetry groups of systems of entangled particles uncovered in [44] It turns out that the Lorentz transformations of signature $(m,n)$ form the symmetry group by which systems of $m$ $n$-dimensional entangled particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity can be understood. Consequently, it is useful to extend special relativity theory by incorporating Lorentz transformation groups of signature $(m,3)$ for all $m\geq 2$. The resulting extended special relativity theory, then, provides not only the symmetry group of the $(1+3)$-dimensional spacetime of particles, but also the symmetry group of the $(m+3)$-dimensional spacetime of systems of $m$ entangled three-dimensional particles, for each $m\geq 2$.
</p>projecteuclid.org/euclid.jgsp/1527127303_20180523220143Wed, 23 May 2018 22:01 EDTDeformation Quantization in the Teaching of Lie Group Representationshttps://projecteuclid.org/euclid.jgsp/1527127304<strong>Alexander J. Balsomo</strong>, <strong>Job A. Nable</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 79--100.</p><p><strong>Abstract:</strong><br/>
We present straightforward and concrete computations of the unitary irreducible representations of the Euclidean motion group ${\rm M}(2)$ employing the methods of deformation quantization. Deformation quantization is a quantization method of classical mechanics and is an autonomous approach to quantum mechanics, arising from the Wigner quasiprobability distributions and Weyl correspondence. We advertise the utility and power of deformation theory in Lie group representations. In implementing this idea, many aspects of the method of orbits are also learned, thus further adding to the mathematical toolkit of the beginning graduate student of physics. Furthermore, the essential unity of many topics in mathematics and physics (such as Lie theory, quantization, functional analysis and symplectic geometry) is witnessed, an aspect seldom encountered in textbooks, in an elementary way.
</p>projecteuclid.org/euclid.jgsp/1527127304_20180523220143Wed, 23 May 2018 22:01 EDTThe Solution to the Three-Body Problem and Some Applicationshttps://projecteuclid.org/euclid.jgsp/1538705200<strong>Ramon Gonzàlez Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 1--61.</p><p><strong>Abstract:</strong><br/>
Here we provide and explain the coordinate transformation according to which every weighted quadratic form of the absolute Cartesian coordinates or velocities of three particles is separable into quadratic terms of the relative and centre-of-mass coordinates or velocities. This solution is applied to define a new set of weighted colour coordinates $YJK$ in the colour space, and also to solve the dynamical system Sun-Earth-Moon. The weighted Laplacian and hence the quantum Hamiltonian operator for a system of three particles are also given in relative coordinates, and applied to calculate the vibrational energy levels of carbon dioxide and the electronic energy of the ground state of the hydrogen-molecule-ion and two-electron atomic systems like the helium atom..
</p>projecteuclid.org/euclid.jgsp/1538705200_20181004220650Thu, 04 Oct 2018 22:06 EDTConformal Changes of Odd-Dimensional Generalized Structureshttps://projecteuclid.org/euclid.jgsp/1538705201<strong>Fereshteh Malek</strong>, <strong>Niloufar H. Kashani</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 63--76.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the integrability of generalized almost contact and contact manifolds after conformal changes. We also study conditions under which the generalized almost contact and normal generalized contact structures, be normal after conformal changes.
</p>projecteuclid.org/euclid.jgsp/1538705201_20181004220650Thu, 04 Oct 2018 22:06 EDTThe Mechanics and Mathematics of Biological Growth by Alain Gorielyhttps://projecteuclid.org/euclid.jgsp/1538705202<strong>Jean-François Ganghoffer</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 77--79.</p>projecteuclid.org/euclid.jgsp/1538705202_20181004220650Thu, 04 Oct 2018 22:06 EDTVasil Valdemarov Tsanovhttps://projecteuclid.org/euclid.jgsp/1538705203<p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 81--106.</p>projecteuclid.org/euclid.jgsp/1538705203_20181004220650Thu, 04 Oct 2018 22:06 EDTLorentz-Invariant Second-Order Tensors and an Irreducible Set of Matriceshttps://projecteuclid.org/euclid.jgsp/1548471825<strong>Mayeul Arminjon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 1--10.</p><p><strong>Abstract:</strong><br/>
We prove that, up to multiplication by a scalar, the Minkowski metric tensor is the only second-order tensor that is Lorentz-invariant. To prove this, we show that a specific set of three $4\times 4$ matrices, made of two rotation matrices plus a Lorentz boost, is irreducible.
</p>projecteuclid.org/euclid.jgsp/1548471825_20190125220351Fri, 25 Jan 2019 22:03 ESTThird-Order Jacobsthal Generalized Quaternionshttps://projecteuclid.org/euclid.jgsp/1548471826<strong>Gamaliel Cerda-Morales</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 11--27.</p><p><strong>Abstract:</strong><br/>
In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.
</p>projecteuclid.org/euclid.jgsp/1548471826_20190125220351Fri, 25 Jan 2019 22:03 ESTSpaces Realized and Non-Realized as Dold-Lashof Classifying Spaceshttps://projecteuclid.org/euclid.jgsp/1548471827<strong>Mohamed Rachid Hilali</strong>, <strong>Abdelhadi Zaim</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 29--56.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a simply connected CW-complex of finite type. Denote by ${\rm Baut}_{1}(X)$ the Dold-Lashof classifying space of fibrations with fiber $X$. \ This paper is a survey about the problem of realizing Dold-Lashof classifying spaces. We will also present some new results: we show that not all rank-two rational $H$-spaces can be realized as ${\rm Baut}_{1}(X)$ for simply connected, rational elliptic space $X$. Moreover, we construct an infinite family of rational spaces $X,$ such that ${\rm Baut}_{1}(X)$ is rationally a finite $H$-space of rank-two (up to rational homotopy type).
</p>projecteuclid.org/euclid.jgsp/1548471827_20190125220351Fri, 25 Jan 2019 22:03 ESTMathematical Model of Elastic Closed Flexible Shells with Nonlocal Shape Deviationshttps://projecteuclid.org/euclid.jgsp/1548471828<strong>Viktor Olevskyi</strong>, <strong>Yuliia Olevska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 57--69.</p><p><strong>Abstract:</strong><br/>
A model of deformation and mechanical stability of a thin-walled shell with geometric deviations, which is close to a circular cylindrical shell, under the action of axial compression and normal pressure is developed. The model uses the scheme of a flexible shell of zero Gaussian curvature with a perturbed edge, which makes it possible to apply the methods of the geometrically nonlinear theory of torso shells.
</p>projecteuclid.org/euclid.jgsp/1548471828_20190125220351Fri, 25 Jan 2019 22:03 ESTExamples of Automorphism Groups of Ind-Varieties of Generalized Flagshttps://projecteuclid.org/euclid.jgsp/1548471829<strong>Ivan Penkov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 71--77.</p><p><strong>Abstract:</strong><br/>
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of the maximal flags indexed by ${\mathbb Z}_{>0}$. We pay special attention to differences with the case of ordinary flag varieties.
</p>projecteuclid.org/euclid.jgsp/1548471829_20190125220351Fri, 25 Jan 2019 22:03 EST2+1-Moulton Configurationhttps://projecteuclid.org/euclid.jgsp/1548471830<strong>Naoko Yoshimi</strong>, <strong>Akira Yoshioka</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 79--95.</p><p><strong>Abstract:</strong><br/>
We pose a new problem of collinear central configuration in Newtonian $n$-body problem. For a given two-body, we ask whether we can add a new body in a way such that i) the configuration of the total three-body is also collinear central with the configuration of the initial two-body being fixed and further ii) the initial two-body keeps its motion without any change during the process. We find three solutions to the above problem. We also consider a similar problem such that while the condition i) is satisfied but by modifying the condition ii) the motion of the initial two-body is not necessarily equal to the original one. We also find explicit solutions to the second problem.
</p>projecteuclid.org/euclid.jgsp/1548471830_20190125220351Fri, 25 Jan 2019 22:03 ESTThe Restricted Three-Body Problem and Holomorphic Curves by Urs Frauenfelder and Otto van Koerthttps://projecteuclid.org/euclid.jgsp/1548471831<strong>Ramon González Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 97--100.</p>projecteuclid.org/euclid.jgsp/1548471831_20190125220351Fri, 25 Jan 2019 22:03 ESTNew Properties of Euclidean Killing Tensors of Rank Twohttps://projecteuclid.org/euclid.jgsp/1556244025<strong>Mircea Crasmareanu</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 1--7.</p><p><strong>Abstract:</strong><br/>
Due to the importance of Killing tensors of rank two in providing quadratic first integrals we point out several algebraic and geometrical features of this class of Killing tensor fields for the two-dimensional Euclidean metric.
</p>projecteuclid.org/euclid.jgsp/1556244025_20190425220035Thu, 25 Apr 2019 22:00 EDTRelations Between Laplace Spectra and Geometric Quantization of Reimannian Symmetric Spaceshttps://projecteuclid.org/euclid.jgsp/1556244026<strong>Dimitar Grantcharov</strong>, <strong>Gueo Grantcharov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 9--28.</p><p><strong>Abstract:</strong><br/>
We consider a modified Kostant-Souriau geometric quantization scheme due to Czyz and Hess for Hamiltonian systems on the cotangent bundles of compact rank-one Riemannian symmetric spaces ({\rm CROSS}). It is used, together with a symplectic reduction process, to relate its energy spectrum to the spectrum of the Laplace-Beltrami operator. Moreover, the corresponding eigenspaces have real dimension equal to the complex dimension of the space of the holomorphic sections of the quantum bundle which is obtained after the quantization. The relation between the two constructions was first noticed by Mladenov and Tsanov for the case of the spheres. In addition to the {\rm CROSS} case, we announce preliminary results related to the case of compact Riemannian symmetric spaces of higher rank.
</p>projecteuclid.org/euclid.jgsp/1556244026_20190425220035Thu, 25 Apr 2019 22:00 EDTOn the Geometry of Orbits of Conformal Vector Fieldshttps://projecteuclid.org/euclid.jgsp/1556244027<strong>Abdigappar Narmanov</strong>, <strong>Eldor Rajabov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 29--39.</p><p><strong>Abstract:</strong><br/>
Geometry of orbit is a subject of many investigations because it has important role in many branches of mathematics such as dynamical systems,control theory. In this paper it is studied geometry of orbits of conformal vector fields. It is shown that orbits of conformal vector fields are integral submanifolds of completely integrable distributions. Also for Euclidean space it is proven that if all orbits have the same dimension they are closed subsets.
</p>projecteuclid.org/euclid.jgsp/1556244027_20190425220035Thu, 25 Apr 2019 22:00 EDTExplicit Description of Some Classes of Non-Bending Surfaceshttps://projecteuclid.org/euclid.jgsp/1556244028<strong>Vladimir I. Pulov</strong>, <strong>Ivaïlo M. Mladenov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 41--71.</p><p><strong>Abstract:</strong><br/>
Here we consider a family of axially symmetric surfaces modeling the shape of thin mechanical shells that are deformable without bending under uniform loading. With the exception of very few surfaces, like the well known right circular cylinder and the sphere, the surfaces of this family have no closed form description in elementary functions. Our main goal is to present their explicit parameterizations including both classes of open and closed families. We distinguish four classes of non-bending surfaces differing by their canonical representations using the normal elliptic integrals and the Jacobian elliptic functions.
</p>projecteuclid.org/euclid.jgsp/1556244028_20190425220035Thu, 25 Apr 2019 22:00 EDTSecant Varieties and Degrees of Invariantshttps://projecteuclid.org/euclid.jgsp/1556244029<strong>Valdemar V. Tsanov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 73--85.</p><p><strong>Abstract:</strong><br/>
The ring of invariant polynomials $\mathbb{C}[V]^G$ over a given finite dimensional representation space $V$ of a connected complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being nonconstructive, the generators and their degrees have remained a subject of interest. \linebreak In this article we determine certain divisors of the degrees of the generators. Also, for irreducible representations, we provide lower bounds for the degrees, determined by the geometric properties of the unique closed projective $G$-orbit $\mathbb{X}$, and more specifically its secant varieties. For a particular class of representations, where the secant varieties are especially well behaved, we exhibit an exact correspondence between the generating invariants and the secant varieties intersecting the semistable locus.
</p>projecteuclid.org/euclid.jgsp/1556244029_20190425220035Thu, 25 Apr 2019 22:00 EDTCharge of D-Braneshttps://projecteuclid.org/euclid.jgsp/1556244030<strong>Andrés Viña</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 51, 81--102.</p><p><strong>Abstract:</strong><br/>
Considering the $D$-branes on a variety $Z$ as the objects of the derived category $D^b(Z)$, we propose a definition for the charge of $D$-branes on not necessarily smooth varieties. We define the charge $Q({\mathcal G})$ of ${\mathcal G}\in D^b(Z)$ as an element of the homology of $Z$, so that the mapping $Q$ is compatible with the pushforward by proper maps between varieties. Given a generic anticanonical hypersurface $Y$ of a toric variety $X$ defined by a reflexive polytope, we express the charge of a line bundle on $Y$ defined by a divisor $D$ of $X$ in terms of intersections of $D$ with cycles determined by the polytope faces.
</p>projecteuclid.org/euclid.jgsp/1556244030_20190425220035Thu, 25 Apr 2019 22:00 EDTModeling of Minimal Surface Based on an Isotropic Bezier Curve of Fifth Orderhttps://projecteuclid.org/euclid.jgsp/1564106593<strong>Nataliia Ausheva</strong>, <strong>Viktor Olevskyi</strong>, <strong>Yuliia Olevska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 52, 1--15.</p><p><strong>Abstract:</strong><br/>
A method is proposed for constructing minimal surfaces based on fifth-order Bezier isotropic curves specified in a vector-parametric form, allowing control of the guide curve and the surface in user mode. The coefficients of the basic quadratic forms were calculated and it was shown that the surfaces would be minimal. An example of a surface constructed by the proposed method is given.
</p>projecteuclid.org/euclid.jgsp/1564106593_20190725220326Thu, 25 Jul 2019 22:03 EDTSome Geometrical Aspects of Einstein, Ricci and Yamabe Solitonshttps://projecteuclid.org/euclid.jgsp/1564106594<strong>Adara M. Blaga</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 52, 17--26.</p><p><strong>Abstract:</strong><br/>
Under certain assumptions, we characterize the almost $\eta$-Einstein, $\eta$-Ricci and $\eta$-Yamabe solitons on a pseudo-Riemannian manifold when the potential vector field of the soliton is infinitezimal harmonic or torse-forming. Moreover, in the second case, if the manifold is Ricci symmetric of constant scalar curvature, then the soliton is completely determined.
</p>projecteuclid.org/euclid.jgsp/1564106594_20190725220326Thu, 25 Jul 2019 22:03 EDTSharp Growth Estimates for Warping Functions in Multiply Warped Product Manifoldshttps://projecteuclid.org/euclid.jgsp/1564106596<strong>Bang-Yen Chen</strong>, <strong>Shihshu Walter Wei</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 52, 27--46.</p><p><strong>Abstract:</strong><br/>
By applying an average method in PDE, we obtain a dichotomy between ``constancy'' and ``infinity'' of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [9], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.
</p>projecteuclid.org/euclid.jgsp/1564106596_20190725220326Thu, 25 Jul 2019 22:03 EDTBi-Hamiltonian Structures on the Tangent Bundle to a Poisson Manifoldhttps://projecteuclid.org/euclid.jgsp/1564106597<strong>Alina Dobrogowska</strong>, <strong>Grzegorz Jakimowicz</strong>, <strong>Karolina Wojciechowicz</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 52, 47--66.</p><p><strong>Abstract:</strong><br/>
In the case when $M$ is equipped with a bi-Hamiltonian structure $(M,\pi_1, \pi_2)$ we show how to construct family of Poisson structures on the tangent bundle $TM$ to a Poisson manifold. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.
</p>projecteuclid.org/euclid.jgsp/1564106597_20190725220326Thu, 25 Jul 2019 22:03 EDTSome Classes of Shapes of the Rotating Liquid Drophttps://projecteuclid.org/euclid.jgsp/1564106598<strong>Vladimir I. Pulov</strong>, <strong>Ivaïlo M. Mladenov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 52, 67--102.</p><p><strong>Abstract:</strong><br/>
The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.
</p>projecteuclid.org/euclid.jgsp/1564106598_20190725220326Thu, 25 Jul 2019 22:03 EDTGeometry of Twisted Sasaki Metrichttps://projecteuclid.org/euclid.jgsp/1572228019<strong>Lakehal Belarbi</strong>, <strong>Hichem Elhendi</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 53, 1--19.</p><p><strong>Abstract:</strong><br/>
Let $(M,g)$ be a $n$-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by $G^{f,h}$ and called twisted Sasaki matric on the tangent bundle $TM$. We studied the geometry of $(TM,G^{f,h})$ by giving a relationships of the curvatures, Einstein structure, scalar and sectional curvatures between $(TM,G^{f,h})$ and $(M,g)$.
</p>projecteuclid.org/euclid.jgsp/1572228019_20191027220024Sun, 27 Oct 2019 22:00 EDTCharacterization and Computation of Matrices of Maximal Trace Over Rotationshttps://projecteuclid.org/euclid.jgsp/1572228020<strong>Javier Bernal</strong>, <strong>Jim Lawrence</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 53, 21--53.</p><p><strong>Abstract:</strong><br/>
Given a $d\times d$ matrix $M$, it is well known that finding a $d\times d$ rotation matrix $U$ that maximizes the trace of $UM$, i.e., that makes $UM$ a matrix of maximal trace over rotation matrices, can be achieved with a method based on the computation of the singular value decomposition (SVD) of~$M$. We characterize $d\times d$ matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for $d=2,3$, we identify alternative ways, other than the SVD, of computing $U$ so that $UM$ is of maximal trace over rotation matrices.
</p>projecteuclid.org/euclid.jgsp/1572228020_20191027220024Sun, 27 Oct 2019 22:00 EDTMechanics of Infinitesimal Test Bodies on Delaunay Surfaces: Spheres and Cylinders as Limits of Unduloids and Their Action-Angle Analysishttps://projecteuclid.org/euclid.jgsp/1572228021<strong>Vasyl Kovalchuk</strong>, <strong>Barbara Gołubowska</strong>, <strong>Ivaïlo Mladenov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 53, 55--83.</p><p><strong>Abstract:</strong><br/>
This paper discusses the motion of infinitesimal gyroscopes along the two-dimensional surfaces of constant mean curvature embedded into the three-dimensional Euclidean space. We have considered the cases of unduloids, spheres, and cylinders for which the corresponding Hamilton-Jacobi equations are written and analyzed with the help of the action-angle variables. Spheres and cylinders are considered as limiting cases of unduloids and the residue analysis was performed which provides the connection between all three action variables and the energy. This has been traced also for the geodetic situations and for two additional classical model potentials.
</p>projecteuclid.org/euclid.jgsp/1572228021_20191027220024Sun, 27 Oct 2019 22:00 EDTEigenvectors of the SO(3,${\mathbb{R}}$) Matriceshttps://projecteuclid.org/euclid.jgsp/1572228022<strong>Amol Sasane</strong>, <strong>Victor Ufnarovski</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 53, 85--102.</p><p><strong>Abstract:</strong><br/>
Let $R=(r_{ij})\in$ SO(3,${\mathbb{R}}$). We give several different proofs of the fact that the vector $$ V:=\Big( \displaystyle \frac{1}{r_{23}+r_{32}}, \displaystyle \frac{1}{r_{13}+r_{31}}, \displaystyle \frac{1}{r_{12}+r_{21}}\Big)^t $$ if it exists, is an eigenvector of $R$ corresponding to the eigenvalue one.
</p>projecteuclid.org/euclid.jgsp/1572228022_20191027220024Sun, 27 Oct 2019 22:00 EDTFermi-Walker Parallel Transport According to Quasi Frame in Three Dimensional Minkowski Spacehttps://projecteuclid.org/euclid.jgsp/1578193215<strong>Nevin Gürbü}z</strong>, <strong>Dae W. Yoon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 54, 1--12.</p><p><strong>Abstract:</strong><br/>
In this paper, we present the Fermi-Walker parallel transport and the generalized Fermi-Walker parallel transport according to quasi frame in three dimensional Minkowski space.
</p>projecteuclid.org/euclid.jgsp/1578193215_20200104220042Sat, 04 Jan 2020 22:00 ESTOn $\Lambda$-Elasticahttps://projecteuclid.org/euclid.jgsp/1578193222<strong>Shigeki Matsutani</strong>, <strong>Hiroshi Nishiguchi</strong>, <strong>Kenji Higashida</strong>, <strong>Akihiro Nakatani</strong>, <strong>Hiroyasu Hamada</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 54, 13--35.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $\phi_0$ and we call the piece-wised elastica {\it{$\Lambda_{\phi_0}$-elastica}} or {\it{$\Lambda$-elastica}}. Such transition appears in the bending beam experiment when an elastic beam is gradually compressed and at some moment suddenly due to the rupture, the shapes of $\Lambda$-elastica appear. We construct a mathematical theory to describe the phenomena and represent the $\Lambda$-elastica in terms of the elliptic $\zeta$-function completely. Using the mathematical theory, we discuss the experimental results from an energetic viewpoint and numerically show the explicit shape of $\Lambda$-elastica. It means that this paper provides a novel investigation on elastica theory with rupture.
</p>projecteuclid.org/euclid.jgsp/1578193222_20200104220042Sat, 04 Jan 2020 22:00 ESTWigner Functions and Weyl Operators on the Euclidean Motion Grouphttps://projecteuclid.org/euclid.jgsp/1578193228<strong>Laarni B. Natividad</strong>, <strong>Job A. Nable</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 54, 37--54.</p><p><strong>Abstract:</strong><br/>
The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl - Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.
</p>projecteuclid.org/euclid.jgsp/1578193228_20200104220042Sat, 04 Jan 2020 22:00 ESTRotating Liquid Drops and Delaunay Surfaceshttps://projecteuclid.org/euclid.jgsp/1578193229<strong>Vladimir I. Pulov</strong>, <strong>Ivaïlo M. Mladenov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 54, 55--78.</p><p><strong>Abstract:</strong><br/>
Here we consider the problem of finding the equilibrium configurations of a rotating liquid drop, paying special attention to the cases when the droplet takes the shape of a Delaunay surface. By making use of the canonical forms of the elliptic integrals and the Jacobian elliptic functions we have derived several explicit parameterizations of the Delaunay surfaces. They are expressed relying on two independant real parameters accounting respectively the size and the shape so that all possible Delaunay surfaces are represented in a unified way.
</p>projecteuclid.org/euclid.jgsp/1578193229_20200104220042Sat, 04 Jan 2020 22:00 ESTOn Analogues of Bäcklund Theorem in Affine Differential Geometry of Surfaceshttps://projecteuclid.org/euclid.jgsp/1578193233<strong>Maria Robaszewska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 54, 79--110.</p><p><strong>Abstract:</strong><br/>
Here we recall the well-known Chern--Terng theorem concerning affine minimal surfaces. After that we formulate some complementary (with transversal fields necessarily not parallel) affine B\" acklund theorem. Next, we describe some geometrical conditions which imply the local symmetry of both induced connections. Finally, we give some necessary and sufficient conditions under which the affine fundamental forms are proportional.
</p>projecteuclid.org/euclid.jgsp/1578193233_20200104220042Sat, 04 Jan 2020 22:00 EST