Pub Date : 2022-01-01DOI: 10.7546/jgsp-64-2022-39-49
Taika Okuda, Akifumi Sako
A construction methods of noncommutative locally symmetric K"ahler manifolds via a deformation quantization with separation of variables was proposed by Sako-Suzuki-Umetsu and Hara-Sako. This construction gives the recurrence relations to determine the star product. These recurrence relations were solved for the case of the arbitrary one-dimensional ones, $N$-dimensional complex space, complex projective space and complex hyperbolic space. For any two-dimensional case, authors found the solution of the recurrence relations. In this paper, we review the solution and make the star product for two-dimensional complex projective space as a concrete example of this solution.
{"title":"Deformaion Quantization with Separation of Variables for Complex Two-Dimensional Locally Symmetric Kähler Manifold","authors":"Taika Okuda, Akifumi Sako","doi":"10.7546/jgsp-64-2022-39-49","DOIUrl":"https://doi.org/10.7546/jgsp-64-2022-39-49","url":null,"abstract":"A construction methods of noncommutative locally symmetric K\"ahler manifolds via a deformation quantization with separation of variables was proposed by Sako-Suzuki-Umetsu and Hara-Sako. This construction gives the recurrence relations to determine the star product. These recurrence relations were solved for the case of the arbitrary one-dimensional ones, $N$-dimensional complex space, complex projective space and complex hyperbolic space. For any two-dimensional case, authors found the solution of the recurrence relations. In this paper, we review the solution and make the star product for two-dimensional complex projective space as a concrete example of this solution.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7546/jgsp-63-2022-39-64
K. Kikuchi, Tsukasa Takeuchi
Ikeda and Sakamoto studied a dynamical control problem called the linear first integral for holonomic dynamical systems, and our proposition proved the same result as theirs in integrability. % Also, a symplectic Haantjes manifolds has been defined by Tempesta and Tondo, which is a characterization of integrable systems using $(1,1)$ tensor fields. We show integrability in dynamical control problems from a geometric point of view by means of a concrete construction of a symplectic Haantjes manifold.
{"title":"Integrability Theorems of Free Systems and Symplectic Haantjes Structures","authors":"K. Kikuchi, Tsukasa Takeuchi","doi":"10.7546/jgsp-63-2022-39-64","DOIUrl":"https://doi.org/10.7546/jgsp-63-2022-39-64","url":null,"abstract":"Ikeda and Sakamoto studied a dynamical control problem called the linear first integral for holonomic dynamical systems, and our proposition proved the same result as theirs in integrability. % Also, a symplectic Haantjes manifolds has been defined by Tempesta and Tondo, which is a characterization of integrable systems using $(1,1)$ tensor fields. We show integrability in dynamical control problems from a geometric point of view by means of a concrete construction of a symplectic Haantjes manifold.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7546/jgsp-64-2022-9-22
Ying-Qiu Gu
By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation $A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, satisfying the associativity and distributive law, which provide a unified, standard, and elegant language for many complex mathematical and physical objects. So, we have one more proof that the hypercomplex numbers are worthy of application in teaching and scientific research.
{"title":"Hypercomplex Numbers and Roots of Algebraic Equation","authors":"Ying-Qiu Gu","doi":"10.7546/jgsp-64-2022-9-22","DOIUrl":"https://doi.org/10.7546/jgsp-64-2022-9-22","url":null,"abstract":"By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation $A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, satisfying the associativity and distributive law, which provide a unified, standard, and elegant language for many complex mathematical and physical objects. So, we have one more proof that the hypercomplex numbers are worthy of application in teaching and scientific research.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7546/jgsp-64-2022-23-28
D. Kamburova, R. Marinov
In this short note we present a new proof of Ekeland's variational principle and Caristi's fixed point theorem using a recently proved constrained variational principle in completely regular topological spaces.
{"title":"Ekeland's Variational Principle and Caristi's Fixed Point Theorem","authors":"D. Kamburova, R. Marinov","doi":"10.7546/jgsp-64-2022-23-28","DOIUrl":"https://doi.org/10.7546/jgsp-64-2022-23-28","url":null,"abstract":"In this short note we present a new proof of Ekeland's variational principle and Caristi's fixed point theorem using a recently proved constrained variational principle in completely regular topological spaces.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7546/jgsp-64-2022-51-65
Binuri Perera, Thanuja Paragoda, Dayal Dharmasena
In this paper we survey Delaunay surfaces in $mathbb{R}^{3}$ spanning two coaxial circles which appear as capillary surfaces supported on different solid supports in the absence of gravity. We classify these surfaces based on contact angles and the geometry of the support. Numerical solutions of the Euler Lagrange equation are provided using numerical methods.
{"title":"A Survey of Delaunay Surfaces with Applications in Capillary Surfaces","authors":"Binuri Perera, Thanuja Paragoda, Dayal Dharmasena","doi":"10.7546/jgsp-64-2022-51-65","DOIUrl":"https://doi.org/10.7546/jgsp-64-2022-51-65","url":null,"abstract":"In this paper we survey Delaunay surfaces in $mathbb{R}^{3}$ spanning two coaxial circles which appear as capillary surfaces supported on different solid supports in the absence of gravity. We classify these surfaces based on contact angles and the geometry of the support. Numerical solutions of the Euler Lagrange equation are provided using numerical methods.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7546/jgsp-63-2022-65-75
I. Mladenov
A plethora of explicit formulas that parameterize any type of the spiric sections are derived from the first principles.
过多的显式公式参数化任何类型的螺旋截面是从第一原理推导出来的。
{"title":"Analytical Descriptions of Perseus Spirics","authors":"I. Mladenov","doi":"10.7546/jgsp-63-2022-65-75","DOIUrl":"https://doi.org/10.7546/jgsp-63-2022-65-75","url":null,"abstract":"A plethora of explicit formulas that parameterize any type of the spiric sections are derived from the first principles.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.7546/jgsp-61-2021-79-104
Tu T. C. Nguyen, V. Le
In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $mathfrak{g}_{5,2}$ given in Table~ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.
{"title":"Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups","authors":"Tu T. C. Nguyen, V. Le","doi":"10.7546/jgsp-61-2021-79-104","DOIUrl":"https://doi.org/10.7546/jgsp-61-2021-79-104","url":null,"abstract":"In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $mathfrak{g}_{5,2}$ given in Table~ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48610500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.7546/jgsp-61-2021-53-78
H. Loumi-Fergane
Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar'e-Cartan form. The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.
{"title":"Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor","authors":"H. Loumi-Fergane","doi":"10.7546/jgsp-61-2021-53-78","DOIUrl":"https://doi.org/10.7546/jgsp-61-2021-53-78","url":null,"abstract":"Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar'e-Cartan form. The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.7546/jgsp-61-2021-41-51
H. Kumara, V. Venkatesha, D. Naik
In this work, we intend to investigate the characteristics of static perfect fluid space-time metrics on almost Kenmotsu manifolds. At first we prove that if a Kenmotsu manifold $M$ is the spatial factor of static perfect fluid space-time then it is $eta$-Einstein. Moreover, if the Reeb vector field $xi$ leaves the scalar curvature invariant, then $M$ is Einstein. Next we consider static perfect fluid space-time on almost Kenmotsu $(kappa,mu)'$-manifolds and give some characteristics under certain conditions.
{"title":"Static Perfect Fluid Space-Time on Almost Kenmotsu Manifolds","authors":"H. Kumara, V. Venkatesha, D. Naik","doi":"10.7546/jgsp-61-2021-41-51","DOIUrl":"https://doi.org/10.7546/jgsp-61-2021-41-51","url":null,"abstract":"In this work, we intend to investigate the characteristics of static perfect fluid space-time metrics on almost Kenmotsu manifolds. At first we prove that if a Kenmotsu manifold $M$ is the spatial factor of static perfect fluid space-time then it is $eta$-Einstein. Moreover, if the Reeb vector field $xi$ leaves the scalar curvature invariant, then $M$ is Einstein. Next we consider static perfect fluid space-time on almost Kenmotsu $(kappa,mu)'$-manifolds and give some characteristics under certain conditions.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44772706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-30DOI: 10.7546/jgsp-61-2021-1-16
Daniele Corradetti
Recent papers contributed revitalizing the study of the exceptional Jordan algebra $mathfrak{h}_{3}(mathbb{O})$ in its relations with the true Standard Model gauge group $mathrm{G}_{SM}$. The absence of complex representations of $mathrm{F}_{4}$ does not allow $Autleft(mathfrak{h}_{3}(mathbb{O})right)$ to be a candidate for any Grand Unified Theories, but the automorphisms of the complexification of this algebra, i.e., $mathfrak{h}_{3}^{mathbb{C}}(mathbb{O})$, are isomorphic to the compact form of $mathrm{E}_{6}$ and similar constructions lead to the gauge group of the minimal left-right symmetric extension of the Standard Model.
{"title":"Complexification of the Exceptional Jordan Algebra and Its Application to Particle Physics","authors":"Daniele Corradetti","doi":"10.7546/jgsp-61-2021-1-16","DOIUrl":"https://doi.org/10.7546/jgsp-61-2021-1-16","url":null,"abstract":"Recent papers contributed revitalizing the study of the exceptional Jordan algebra $mathfrak{h}_{3}(mathbb{O})$ in its relations with the true Standard Model gauge group $mathrm{G}_{SM}$. The absence of complex representations of $mathrm{F}_{4}$ does not allow $Autleft(mathfrak{h}_{3}(mathbb{O})right)$ to be a candidate for any Grand Unified Theories, but the automorphisms of the complexification of this algebra, i.e., $mathfrak{h}_{3}^{mathbb{C}}(mathbb{O})$, are isomorphic to the compact form of $mathrm{E}_{6}$ and similar constructions lead to the gauge group of the minimal left-right symmetric extension of the Standard Model.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45739733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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