IIn this paper, we examine PNMCV-MCGL biconservative submanifold in a Minkowski space $mathbb{E}_1^{n+2}$ with nondiagonalizable shape operator, where PNMCV-MCGL submanifold denotes a submanifold with parallel normalized mean curvature vector and the mean curvature whose gradient is lightlike ($langlenabla H,nabla Hrangle=0$). We obtain some conditions about connection forms, principal curvatures and some results about them. Then we use them to obtain a classification of such submanifolds. Finally, we showed that there is no biconservative such submanifold in Minkowski space of arbitrary dimension.
{"title":"A Classification of Parallel Normalized Biconservative Submanifold in the Minkowski Space in Arbitrary Dimension","authors":"Aykut KAYHAN","doi":"10.36890/iejg.1263203","DOIUrl":"https://doi.org/10.36890/iejg.1263203","url":null,"abstract":"IIn this paper, we examine PNMCV-MCGL biconservative submanifold in a Minkowski space $mathbb{E}_1^{n+2}$ with nondiagonalizable shape operator, where PNMCV-MCGL submanifold denotes a submanifold with parallel normalized mean curvature vector and the mean curvature whose gradient is lightlike ($langlenabla H,nabla Hrangle=0$). We obtain some conditions about connection forms, principal curvatures and some results about them. Then we use them to obtain a classification of such submanifolds. Finally, we showed that there is no biconservative such submanifold in Minkowski space of arbitrary dimension.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136134485","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 consider the stratification of orbit spaces as defined by the local diffeomorphism action, known as the Klein stratification. We demonstrate that the Klein strata on orbit spaces with isolated point singularities are precisely the union of points where the space has the same dimension.
{"title":"Klein Stratification of Orbit spaces","authors":"Serap GÜRER","doi":"10.36890/iejg.1364214","DOIUrl":"https://doi.org/10.36890/iejg.1364214","url":null,"abstract":"We consider the stratification of orbit spaces as defined by the local diffeomorphism action, known as the Klein stratification. We demonstrate that the Klein strata on orbit spaces with isolated point singularities are precisely the union of points where the space has the same dimension.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"59 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136133648","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}
Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2times R$, $ S^2times R$, $Nil$, $widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.
{"title":"Classical notions and problems in Thurston geometries","authors":"Jenő SZİRMAİ","doi":"10.36890/iejg.1221802","DOIUrl":"https://doi.org/10.36890/iejg.1221802","url":null,"abstract":"Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2times R$, $ S^2times R$, $Nil$, $widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136134482","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}
Jin introduced a non-symmetric metric connection, called an {it $(ell,m)$-type metric connection} cite{Jin1, Jin2}. There are two examples of $(ell, m)$-type: a semi-symmetric metric connection when ${ell}=1$ and $m=0$ and a quater-symmetric connection for ${ell}=0$ and $m=1$ . Our purpose is to investigate lightlike hypersurfaces of an indefinite (complex) Kaehler manifolds with an $(ell,m)$-type metric connection under the tangent characteristic vector field on such hypersurfaces.
{"title":"Lightlike hypersurfaces of an indefinite Kaehler manifold with an $(ell,,m)$-type metric connection","authors":"Dae Ho JİN, Chul Woo LEE, Jae Won LEE","doi":"10.36890/iejg.1264249","DOIUrl":"https://doi.org/10.36890/iejg.1264249","url":null,"abstract":"Jin introduced a non-symmetric metric connection, called an {it $(ell,m)$-type metric connection} cite{Jin1, Jin2}. There are two examples of $(ell, m)$-type: a semi-symmetric metric connection when ${ell}=1$ and $m=0$ and a quater-symmetric connection for ${ell}=0$ and $m=1$ . Our purpose is to investigate lightlike hypersurfaces of an indefinite (complex) Kaehler manifolds with an $(ell,m)$-type metric connection under the tangent characteristic vector field on such hypersurfaces.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"46 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136133890","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}
Miroslava PETROVİĆ-TORGAŠEV, Ryszard DESZCZ, Małgorzata GŁOGOWSKA, Marian HOTLOŚ, Georges ZAFİNDRATAFA
The derivation-commutator $R cdot C - C cdot R$ of a semi-Riemannian manifold $(M,g)$, $dim M geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.
微分对易子 $R cdot C - C cdot R$ 半黎曼流形的 $(M,g)$， $dim M geq 4$由它的黎曼-克里斯托费尔曲率张量构成 $R$ 和Weyl共形曲率张量 $C$，在某些假设下，可以表示为的线性组合 $(0,6)$-立花张量 $Q(A,T)$，其中 $A$ 是对称的 $(0,2)$-张量和 $T$ 广义曲率张量。这些条件构成了广义爱因斯坦度规条件的一类。在这篇综述文章中，我们给出了最近关于流形和子流形，特别是超曲面，满足这些条件的结果。
{"title":"ON SEMI-RIEMANNIAN MANIFOLDS SATISFYING SOME GENERALIZED EINSTEIN METRIC CONDITIONS","authors":"Miroslava PETROVİĆ-TORGAŠEV, Ryszard DESZCZ, Małgorzata GŁOGOWSKA, Marian HOTLOŚ, Georges ZAFİNDRATAFA","doi":"10.36890/iejg.1323352","DOIUrl":"https://doi.org/10.36890/iejg.1323352","url":null,"abstract":"The derivation-commutator $R cdot C - C cdot R$ of a semi-Riemannian manifold $(M,g)$, $dim M geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"46 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136133891","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}
Noura DJELLALI, Abdelbasset HASNİ, Ahmed MOHAMMED CHERİF, Mohamed BELKHELFA
In this paper, we give a classification of Codazzi hypersurfaces in a Lie group $(Nil^{4},widetilde g)$. We also give a characterization of a class of minimal hypersurfaces in $(Nil^{4},widetilde g)$ with an example of a minimal surface in this class.
{"title":"Classification of Codazzi and note on minimal hypersurfaces in $Nil^{4}$","authors":"Noura DJELLALI, Abdelbasset HASNİ, Ahmed MOHAMMED CHERİF, Mohamed BELKHELFA","doi":"10.36890/iejg.1256112","DOIUrl":"https://doi.org/10.36890/iejg.1256112","url":null,"abstract":"In this paper, we give a classification of Codazzi hypersurfaces in a Lie group $(Nil^{4},widetilde g)$. We also give a characterization of a class of minimal hypersurfaces in $(Nil^{4},widetilde g)$ with an example of a minimal surface in this class.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136133810","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 an algebraic proof of the Pentagon Theorem. The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension. In particular, the theorem holds in every finite Miquelian plane. The arguments also reveal that the five concyclic points in the Pentagon Theorem are either pairwise distinct or identical to one single point. In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.
{"title":"The Pentagon Theorem in Miquelian Möbius planes","authors":"Lorenz HALBEISEN, Norbert HUNGERBÜHLER, Vanessa LOUREİRO","doi":"10.36890/iejg.1255469","DOIUrl":"https://doi.org/10.36890/iejg.1255469","url":null,"abstract":"We give an algebraic proof of the Pentagon Theorem. The proof works in all Miquelian Möbius planes obtained from a separable quadratic field extension. In particular, the theorem holds in every finite Miquelian plane. The arguments also reveal that the five concyclic points in the Pentagon Theorem are either pairwise distinct or identical to one single point. In addition we identify five additional quintuples of points in the pentagon configuration which are concyclic.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903850","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 present paper is devoted to 4-dimentional Hermitain manifold. We give a new necessary and sufficient condition of integrability and we introduce a new class of locally conformal K"ahler manifolds that we consider a twin of the Vaisman ones. Then, some basic properties of this class is discussed, also the existence of such manifolds is shown with concrete examples.
{"title":"On four dimensional Hermitian manifolds","authors":"Beldjilali GHERİCİ","doi":"10.36890/iejg.1258996","DOIUrl":"https://doi.org/10.36890/iejg.1258996","url":null,"abstract":"The present paper is devoted to 4-dimentional Hermitain manifold. We give a new necessary and sufficient condition of integrability and we introduce a new class of locally conformal K\"ahler manifolds that we consider a twin of the Vaisman ones. Then, some basic properties of this class is discussed, also the existence of such manifolds is shown with concrete examples.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135098091","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 present a Weierstrass-type representation formula which locally represents every regular two-dimensional lightlike surface in Lorentz-Minkowski 4-Space $mathbb{M}^4$ by three dual functions $(rho,f,g)$ and generalizes the representation for regular lightlike surfaces in $mathbb{M}^3$. We give necessary and sufficient conditions on the functions $rho$, $f$, $g$ for the surface to be minimal, ruled or $l$-minimal. For ruled lightlike surfaces, we give necessary and sufficient conditions for the representation itself to be ruled. Furthermore, we give a result on totally geodesic half-lightlike surfaces which holds only in $mathbb{M}^4$.
给出了一个weierstrass型的表示公式，该公式用三个对偶函数$(rho,f,g)$局部表示了Lorentz-Minkowski 4- space $mathbb{M}^4$中的每一个正则二维类光曲面，并推广了$mathbb{M}^3$中的正则类光曲面的表示。给出了函数$rho$， $f$， $g$使曲面极小、直棱或$l$-极小的充分必要条件。对于直纹类光曲面，给出了表象本身被直纹的充分必要条件。此外，我们给出了只在$mathbb{M}^4$中成立的全测地线半类光曲面的结果。
{"title":"Weierstrass Representation of Lightlike Surfaces in Lorentz-Minkowski 4-Space","authors":"Davor DEVALD, Z. MİLİN SİPUS","doi":"10.36890/iejg.1272924","DOIUrl":"https://doi.org/10.36890/iejg.1272924","url":null,"abstract":"We present a Weierstrass-type representation formula which locally represents every regular two-dimensional lightlike surface in Lorentz-Minkowski 4-Space $mathbb{M}^4$ by three dual functions $(rho,f,g)$ and generalizes the representation for regular lightlike surfaces in $mathbb{M}^3$. We give necessary and sufficient conditions on the functions $rho$, $f$, $g$ for the surface to be minimal, ruled or $l$-minimal. For ruled lightlike surfaces, we give necessary and sufficient conditions for the representation itself to be ruled. Furthermore, we give a result on totally geodesic half-lightlike surfaces which holds only in $mathbb{M}^4$.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135563923","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 study translation surfaces generated by spherical indicatrices of timelike curves in Minkowski 3-space and find necessary and sufficient conditions for the translation surfaces to be flat or minimal. Further, we obtain necessary and sufficient conditions for generating curves of the translation surfaces to be geodesic, asymptotic line and line of curvature. Finally for such translation surfaces we obtain the axis when they are constant angle surfaces.
{"title":"Some Characterizations of Translation Surface Generated by Spherical Indicatrices of Timelike Curves in Minkowski 3-space","authors":"A. Yadav, Ajay Kumar Yadav","doi":"10.36890/iejg.1178802","DOIUrl":"https://doi.org/10.36890/iejg.1178802","url":null,"abstract":"In this paper, we study translation surfaces generated by spherical indicatrices of timelike curves in Minkowski 3-space and find necessary and sufficient conditions for the translation surfaces to be flat or minimal. Further, we obtain necessary and sufficient conditions for generating curves of the translation surfaces to be geodesic, asymptotic line and line of curvature. \u0000 Finally for such translation surfaces we obtain the axis when they are constant angle surfaces.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45936638","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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