In this note, we present a biographical sketch of the life and academic contributions of late Professor Krishan L. Duggal. His contributions span from Riemannian and Lorentzian geometries of manifolds with various structural groups of the tangent bundle, Lightlike curves and submanifolds, Cauchy-Riemann geometry, Symmetries of semi-Riemannian manifolds, to Killing horizons. In particular, his approach to the study of lightlike submanifolds is remarkable and drawn considerable interest of many geometers.
{"title":"Professor Krishan L. Duggal: A Biographical Note","authors":"R. Sharma, B. Şahin","doi":"10.36890/iejg.1253398","DOIUrl":"https://doi.org/10.36890/iejg.1253398","url":null,"abstract":"In this note, we present a biographical sketch of the life and academic contributions of late Professor Krishan L. Duggal. His contributions span from Riemannian and Lorentzian geometries of manifolds with various structural groups of the tangent bundle, Lightlike curves and submanifolds, Cauchy-Riemann geometry, Symmetries of semi-Riemannian manifolds, to Killing horizons. In particular, his approach to the study of lightlike submanifolds is remarkable and drawn considerable interest of many geometers.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46210897","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}
R. Deszcz, M. Glogowska, Marian Hotlo's, Miroslava Petrovi'c-Torgavsev, G. Zafindratafa
For any semi-Riemannian manifold (M, g) we define some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.
{"title":"A Note on Some Generalized Curvature Tensor","authors":"R. Deszcz, M. Glogowska, Marian Hotlo's, Miroslava Petrovi'c-Torgavsev, G. Zafindratafa","doi":"10.36890/iejg.1273631","DOIUrl":"https://doi.org/10.36890/iejg.1273631","url":null,"abstract":"For any semi-Riemannian manifold (M, g) we define some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47786086","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}
The goal of the present paper is to analyze some geometric features of Clairaut pointwise slant submersions whose total manifold is a locally product Riemannian manifold. We describe Clairaut pointwise slant submersions from locally product Riemannian manifold onto a Riemannian manifold. We study pointwise slant submersions by providing a consequent which defines the geodesics on the total space of this type submersions. We also give a non-trivial example of the Clairaut pointwise slant submersions whose total manifolds are locally product Riemannian.
{"title":"Clairaut pointwise slant submersion from locally product Riemannian manifolds","authors":"Murat Polat","doi":"10.36890/iejg.1108703","DOIUrl":"https://doi.org/10.36890/iejg.1108703","url":null,"abstract":"The goal of the present paper is to analyze some geometric features of Clairaut pointwise slant submersions whose total manifold is a locally product Riemannian manifold. We describe Clairaut pointwise slant submersions from locally product Riemannian manifold onto a Riemannian manifold. We study pointwise slant submersions by providing a consequent which defines the geodesics on the total space of this type submersions. We also give a non-trivial example of the Clairaut pointwise slant submersions whose total manifolds are locally product Riemannian.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49225571","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}
In this paper, we examine the proper semi-slant pseudo-Riemannian submersions in para-Kaehler geometry and prove some fundamental results on such submersions. In particular we obtain curvature relations in para-Kaehler space forms. Moreover, we provide examples of proper semi-slant pseudo-Riemannian submersions.
{"title":"Proper Semi-Slant Pseudo-Riemannian Submersions in Para-Kaehler Geometry","authors":"Esra BAŞARIR NOYAN, Yılmaz Gündüzalp","doi":"10.36890/iejg.1033345","DOIUrl":"https://doi.org/10.36890/iejg.1033345","url":null,"abstract":"In this paper, we examine the proper semi-slant pseudo-Riemannian submersions in para-Kaehler geometry and prove some fundamental results on such submersions. In particular we obtain curvature relations in para-Kaehler space forms. Moreover, we provide examples of proper semi-slant pseudo-Riemannian submersions.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42076080","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}
The object of the upcoming article is to characterize paracontact metric manifolds conceding $m$-quasi Einstein metric. First we establish that if the metric $g$ in a $K$-paracontact manifold is the $m$-quasi Einstein metric, then the manifold is of constant scalar curvature. Furthermore, we classify $(k,mu)$-paracontact metric manifolds whose metric is $m$-quasi Einstein metric. Finally, we construct a non-trivial example of such a manifold.
{"title":"m-quasi Einstein Metric and Paracontact Geometry","authors":"K. De, U. De, F. Mofarreh","doi":"10.36890/iejg.1100147","DOIUrl":"https://doi.org/10.36890/iejg.1100147","url":null,"abstract":"The object of the upcoming article is to characterize paracontact metric manifolds conceding $m$-quasi Einstein metric. First we establish that if the metric $g$ in a $K$-paracontact manifold is the $m$-quasi Einstein metric, then the manifold is of constant scalar curvature. Furthermore, we classify $(k,mu)$-paracontact metric manifolds whose metric is $m$-quasi Einstein metric. Finally, we construct a non-trivial example of such a manifold.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47504431","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}
In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $le 2$. As a consequence, the eight homogeneous Thurston 3-geometries are realized by tetrahedron manifolds of Heegaard genus $le 2$. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.
{"title":"All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds","authors":"A. Cavicchioli, F. Spaggiari","doi":"10.36890/iejg.1102753","DOIUrl":"https://doi.org/10.36890/iejg.1102753","url":null,"abstract":"In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $le 2$. As a consequence, the eight homogeneous Thurston 3-geometries are realized by tetrahedron manifolds of Heegaard genus $le 2$. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47734161","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}
This study deals with hyperbolic number forms of Euler-Savary Equation (ESE) to find either the four special points on the pole ray. While obtaining the hyperbolic ESE forms, one-parameter planar motion is considered according to the osculating circles contacting at three infinitesimally close points. This approach with the hyperbolic number method gives more detailed information than the traditional method. As a final part, examples are given to show the utility of the practical way in the application.
{"title":"Hyperbolic Number Forms of Euler-Savary Equation","authors":"Duygu Çağlar, N. Gürses","doi":"10.36890/iejg.1127959","DOIUrl":"https://doi.org/10.36890/iejg.1127959","url":null,"abstract":"This study deals with hyperbolic number forms of Euler-Savary Equation (ESE) to find either the four special points on the pole ray. While obtaining the hyperbolic ESE forms, one-parameter planar motion is considered according to the osculating circles contacting at three infinitesimally close points. This approach with the hyperbolic number method gives more detailed information than the traditional method. As a final part, examples are given to show the utility of the practical way in the application.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43425426","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}
Let E_{3} be the 3-dimensional Euclidean space and S be a set such that it has at least two elements. A definition of an S-parametric figure in E_{3} and a definition of a motion of an S-parametric figure in E_{3} are given. Complete systems of G-invariants of a parametric figure in E_{3} for fundamental groups of transformations of E_{3} have obtained. A complete system of G-invariants of a motion of a parametric figure in E_{3} for the Galileo groups Gal_{1}(3,R), Gal^{+}_{1}(3,R) of transformations of E_{3} have obtained.
{"title":"Complete systems of Galileo invariants of a motion of parametric figure in the three dimensional Euclidean space","authors":"D. Khadjiev, İdris Ören, Gayrat Beshimov","doi":"10.36890/iejg.1091348","DOIUrl":"https://doi.org/10.36890/iejg.1091348","url":null,"abstract":"Let E_{3} be the 3-dimensional Euclidean space and S be a set such that it has at least two elements. A definition of an S-parametric figure in E_{3} and a definition of a motion of an S-parametric figure in E_{3} are given. Complete systems of G-invariants of a parametric figure in E_{3} for fundamental groups of transformations of E_{3} have obtained. A complete system of G-invariants of a motion of a parametric figure in E_{3} for the Galileo groups Gal_{1}(3,R), Gal^{+}_{1}(3,R) of transformations of E_{3} have obtained.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46161081","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}
A surface of revolution is a surface that can be generated by rotating a planar curve (the directrix) around a straight line (the axis) in the same plane. Using the mathematics of quaternions, we provide a parametric equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix. Examples are provided to illustrate our algorithm.
{"title":"DETERMINE WHEN A PARAMETRIC SURFACE IS A SURFACE OF REVOLUTION","authors":"Haohao Wang, Jerzy Wojdylo","doi":"10.36890/iejg.1064089","DOIUrl":"https://doi.org/10.36890/iejg.1064089","url":null,"abstract":"A surface of revolution is a surface that can be generated by rotating a planar curve (the directrix)\u0000around a straight line (the axis) in the same plane. Using the mathematics of quaternions, we provide a parametric\u0000equation of a surface of revolution generated by rotating a directrix about an axis by quaternion multiplication\u0000of the parametric representations of the directrix curve and the line of axis. Then, we describe an algorithm\u0000to determine whether a parametric surface is a surface of revolution, and identify the axis and the directrix.\u0000Examples are provided to illustrate our algorithm.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47077991","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}
We give necessary and sufficient conditions, both algebraic and geometric, for a quadrilateral to be the level set of the sum of the distances to m ≥ 2 different lines.
我们给出了四边形是到m≥2条不同直线的距离之和的水平集的代数和几何充要条件。
{"title":"Quadrilaterals as Geometric Loci","authors":"L. Halbeisen, N. Hungerbühler, Juan Läuchli","doi":"10.36890/iejg.1062741","DOIUrl":"https://doi.org/10.36890/iejg.1062741","url":null,"abstract":"We give necessary and sufficient conditions, both algebraic and geometric, for a quadrilateral to be the level set of the sum of the distances to m ≥ 2 different lines.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44084463","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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