Pub Date : 2019-01-01DOI: 10.7546/jgsp-52-2019-67-102
V. Pulov, I. Mladenov
Presented by Ivaïlo Mladenov Abstract. 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. MSC : 53A04, 53A05, 53A10, 53B50, 33E05, 53C22, 76B45, 76D45
{"title":"Some Classes of Shapes of the Rotating Liquid Drop","authors":"V. Pulov, I. Mladenov","doi":"10.7546/jgsp-52-2019-67-102","DOIUrl":"https://doi.org/10.7546/jgsp-52-2019-67-102","url":null,"abstract":"Presented by Ivaïlo Mladenov Abstract. 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. MSC : 53A04, 53A05, 53A10, 53B50, 33E05, 53C22, 76B45, 76D45","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196614","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 : 2019-01-01DOI: 10.7546/jgsp-51-2019-1-7
M. Crasmareanu
A symmetric tensor field on a Riemannian manifold is called a Killing tensor field if the symmetric part of its covariant derivative is equal to zero. There exists a well-known bijection between Killing tensor fields and conserved quantities of the geodesic flow which depend polynomially on the momentum variables. In particular, Killing tensors of rank (or valence) two yields quadratic first integrals and we discuss some aspects of this process in Crasmareanu [7] from a dynamical point of view. Some classes of physical examples associated with the Euclidean 2D metric are provided in Crasmareanu and Baleanu [8].
{"title":"New Properties of Euclidean Killing Tensors of Rank Two","authors":"M. Crasmareanu","doi":"10.7546/jgsp-51-2019-1-7","DOIUrl":"https://doi.org/10.7546/jgsp-51-2019-1-7","url":null,"abstract":"A symmetric tensor field on a Riemannian manifold is called a Killing tensor field if the symmetric part of its covariant derivative is equal to zero. There exists a well-known bijection between Killing tensor fields and conserved quantities of the geodesic flow which depend polynomially on the momentum variables. In particular, Killing tensors of rank (or valence) two yields quadratic first integrals and we discuss some aspects of this process in Crasmareanu [7] from a dynamical point of view. Some classes of physical examples associated with the Euclidean 2D metric are provided in Crasmareanu and Baleanu [8].","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71195967","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 : 2019-01-01DOI: 10.7546/jgsp-54-2019-55-78
V. Pulov, I. Mladenov
Presented by Ivaïlo M. Mladenov Abstract. 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 independent real parameters accounting respectively the size and the shape so that all possible Delaunay surfaces are represented in a unified way. MSC : 53A04, 53A05, 53A10, 53B50, 33E05, 76B45, 76D45
由Ivaïlo M. Mladenov提出摘要。这里我们考虑寻找一个旋转液滴的平衡构型的问题，特别注意液滴呈德劳奈表面形状的情况。利用椭圆积分的标准形式和雅可比椭圆函数，导出了德劳内曲面的几个显式参数化。它们的表示依赖于两个独立的实参数，分别表示大小和形状，以便所有可能的德劳内曲面都以统一的方式表示。MSC: 53a04, 53a05, 53a10, 53b50, 33e05, 76b45, 76d45
{"title":"Rotating Liquid Drops and Delaunay Surfaces","authors":"V. Pulov, I. Mladenov","doi":"10.7546/jgsp-54-2019-55-78","DOIUrl":"https://doi.org/10.7546/jgsp-54-2019-55-78","url":null,"abstract":"Presented by Ivaïlo M. Mladenov Abstract. 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 independent real parameters accounting respectively the size and the shape so that all possible Delaunay surfaces are represented in a unified way. MSC : 53A04, 53A05, 53A10, 53B50, 33E05, 76B45, 76D45","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196589","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 : 2019-01-01DOI: 10.7546/jgsp-51-2019-41-71
V. Pulov, I. Mladenov
{"title":"Explicit Description of Some Classes of Non-Bending Surfaces","authors":"V. Pulov, I. Mladenov","doi":"10.7546/jgsp-51-2019-41-71","DOIUrl":"https://doi.org/10.7546/jgsp-51-2019-41-71","url":null,"abstract":"","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196155","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 : 2018-11-29DOI: 10.7546/jgsp-51-2019-73-85
V. Tsanov
The ring of invariant polynomials ${mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a 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. 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.
{"title":"Secant Varieties and Degrees of Invariants","authors":"V. Tsanov","doi":"10.7546/jgsp-51-2019-73-85","DOIUrl":"https://doi.org/10.7546/jgsp-51-2019-73-85","url":null,"abstract":"The ring of invariant polynomials ${mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a 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. 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.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49010442","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 : 2018-09-15DOI: 10.7546/JGSP-52-2019-27-46
Bang‐Yen Chen, S. Wei
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_1times_{f_2} N_2 times cdots times _{f_k} N_k, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], 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.
{"title":"Sharp Growth Estimates for Warping Functions in Multiply Warped Product Manifolds","authors":"Bang‐Yen Chen, S. Wei","doi":"10.7546/JGSP-52-2019-27-46","DOIUrl":"https://doi.org/10.7546/JGSP-52-2019-27-46","url":null,"abstract":"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_1times_{f_2} N_2 times cdots times _{f_k} N_k, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], 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.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48198486","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 : 2017-03-30DOI: 10.7546/jgsp-55-2020-21-37
Guillermo Capobianco, W. Reartes
In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of $S^1$, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.
{"title":"Holomorphic Path Integrals in Tangent Space for Flat Manifolds","authors":"Guillermo Capobianco, W. Reartes","doi":"10.7546/jgsp-55-2020-21-37","DOIUrl":"https://doi.org/10.7546/jgsp-55-2020-21-37","url":null,"abstract":"In this paper we study the quantum evolution in a flat Riemannian manifold. The holomorphic functions are defined on the cotangent bundle of this manifold. We construct Hilbert spaces of holomorphic functions in which the scalar product is defined using the exponential map. The quantum evolution is proposed by means of an infinitesimal propagator and the holomorphic Feynman integral is developed via the exponential map. The integration corresponding to each step of the Feynman integral is performed in the tangent space. Moreover, in the case of $S^1$, the method proposed in this paper naturally takes into account paths that must be included in the development of the corresponding Feynman integral.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2017-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42245095","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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