In this article, we find an analytical characteristic of the type of a line and derive the formulae for calculating the coordinates of the midpoints and quasi-midpoints of elliptic, hyperbolic, and parabolic segments in an extended hyperbolic space $H^3$ in the frame of the first type. The space $H^3$ we consider in the Cayley,--,Klein projective model as a projective three-dimensional space with an oval quadric $gamma$ fixed in it.
{"title":"Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space","authors":"L. Romakina","doi":"10.36890/iejg.1270550","DOIUrl":"https://doi.org/10.36890/iejg.1270550","url":null,"abstract":"In this article, we find an analytical characteristic of the type of a line and derive the formulae for calculating the coordinates of the midpoints and quasi-midpoints of elliptic, hyperbolic, and parabolic segments in an extended hyperbolic space $H^3$ in the frame of the first type. The space $H^3$ we consider in the Cayley,--,Klein projective model as a projective three-dimensional space with an oval quadric $gamma$ fixed in it.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48321375","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 introduce a new type of curvature function and associated evolute curve for a given curve in the hyperboloid model of plane hyperbolic geometry. A special attention is devoted to the examples, particularly to a horocycle provided by the null Lorentzian rotation.
{"title":"The Flow-geodesic Curvature and the Flow-evolute of Hyperbolic Plane Curves","authors":"M. Crasmareanu","doi":"10.36890/iejg.1229215","DOIUrl":"https://doi.org/10.36890/iejg.1229215","url":null,"abstract":"We introduce a new type of curvature function and associated evolute curve for a given curve in the hyperboloid model of plane hyperbolic geometry. A special attention is devoted to the examples, particularly to a horocycle provided by the null Lorentzian rotation.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48721596","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 work we study vertical graph surfaces invariant by parabolic screw motions with pitch $ell >0$ and constant Gaussian curvature or constant extrinsic curvature in the product space $mathbb H^2 times mathbb R$. In particular, we determine flat and extrinsically flat graph surfaces in $mathbb H^2 times mathbb R$. We also obtain complete and non-complete vertical graph surfaces in $mathbb H^2 times mathbb R$ with negative constant Gaussian curvature and zero extrinsic curvature.
{"title":"Graph Surfaces Invariant by Parabolic screw Motions with Constant Curvature in $ : mathbb H^2 times mathbb R$","authors":"U. Dursun","doi":"10.36890/iejg.1231759","DOIUrl":"https://doi.org/10.36890/iejg.1231759","url":null,"abstract":"In this work we study vertical graph surfaces invariant by parabolic screw motions with pitch $ell >0$ and constant Gaussian curvature or constant extrinsic curvature in the product space $mathbb H^2 times mathbb R$. In particular, we determine flat and extrinsically flat graph surfaces in $mathbb H^2 times mathbb R$. We also obtain complete and non-complete vertical graph surfaces in $mathbb H^2 times mathbb R$ with negative constant Gaussian curvature and zero extrinsic curvature.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43746562","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 class of isotropic submanifolds in pseudo-Riemannian manifolds is a distinguished family of submanifolds; they have been studied by several authors. In this article we establish Chen inequalities for isotropic immersions. An example of an isotropic immersion for which the equality case in the Chen first inequality holds is given.
{"title":"Chen Inequalities for Isotropic Submanifolds in Pseudo-Riemannian Space Forms","authors":"Marius Mi̇rea","doi":"10.36890/iejg.1259890","DOIUrl":"https://doi.org/10.36890/iejg.1259890","url":null,"abstract":"The class of isotropic submanifolds in pseudo-Riemannian manifolds is a distinguished family\u0000of submanifolds; they have been studied by several authors. In this article we establish Chen\u0000inequalities for isotropic immersions. An example of an isotropic immersion for which the equality\u0000case in the Chen first inequality holds is given.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41655956","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 the present paper, we prove that if the metric of a three dimensional almost Kenmotsu manifold with $textbf{Q}phi=phi textbf{Q}$ whose scalar curvature remains invariant under the chracterstic vector field $zeta$, admits a non-trivial Yamabe solitons, then the manifold is of constant sectional curvature or the manifold is Ricci simple.
{"title":"A Note on Yamabe Solitons on 3-dimensional Almost Kenmotsu Manifolds with $textbf{Q}phi=phi textbf{Q}$","authors":"G. Ghosh","doi":"10.36890/iejg.1239222","DOIUrl":"https://doi.org/10.36890/iejg.1239222","url":null,"abstract":"In the present paper, we prove that if the metric of a three dimensional almost Kenmotsu manifold with $textbf{Q}phi=phi textbf{Q}$ whose scalar curvature remains invariant under the chracterstic vector field $zeta$, admits a non-trivial Yamabe solitons, then the manifold is of constant sectional curvature or the manifold is Ricci simple.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46817258","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 magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.
我们考虑双曲三维空间中对应于杀伤磁场的磁曲线。
{"title":"Killing Magnetic Curves in $mathbb{H}^{3}$","authors":"Zlatko Erjavec, J. Inoguchi","doi":"10.36890/iejg.1243521","DOIUrl":"https://doi.org/10.36890/iejg.1243521","url":null,"abstract":"We consider magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45926085","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 work, we handle timelike Aminov surfaces in E_1^4 with respect to having pointwise one type Gauss map. Firstly, we get the laplace of Gauss map of this type of surface. Then, we obtain that there is no timelike Aminov surface having harmonic Gauss map and also pointwise one type Gauss map of first kind in Minkowski 4-space. Further, we yield the conditions of having pointwise one type Gauss map of second kind.
{"title":"An Investigation of Timelike Aminov Surface with respect to its Gauss Map in Minkowski Space-time","authors":"S. Büyükkütük","doi":"10.36890/iejg.1195178","DOIUrl":"https://doi.org/10.36890/iejg.1195178","url":null,"abstract":"In this work, we handle timelike Aminov surfaces in E_1^4 with respect to having pointwise one type Gauss map. Firstly, we get the laplace of Gauss map of this type of surface. Then, we obtain that there is no timelike Aminov surface having harmonic Gauss map and also pointwise one type Gauss map of first kind in Minkowski 4-space. Further, we yield the conditions of having pointwise one type Gauss map of second kind.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47360477","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 theory of polyhedra and the geometric methods associated with it are interesting not only in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author’s work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other. Examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and isometric on sections. In this paper, we prove the positive definiteness of the found invariant. Further, conditional external curvature is introduced for “basic” sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.
{"title":"Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature","authors":"Anvarjon Sharipov, Mukhamedali Keunimjaev","doi":"10.36890/iejg.1246589","DOIUrl":"https://doi.org/10.36890/iejg.1246589","url":null,"abstract":"The theory of polyhedra and the geometric methods associated with it are interesting not only in their own right but also have a wide outlet in the general theory of surfaces. Certainly, it is only sometimes possible to obtain the corresponding theorem on surfaces from the theorem on polyhedra by passing to the limit. Still, the theorems on polyhedra give directions for searching for the related theorems on surfaces. In the case of polyhedra, the elementary-geometric basis of more general results is revealed. In the present paper, we study polyhedra of a particular class, i.e., without edges and reference planes perpendicular to a given direction. This work is a logical continuation of the author’s work, in which an invariant of convex polyhedra isometric on sections was found. The concept of isometry of surfaces and the concept of isometry on sections of surfaces differ from each other. Examples of isometric surfaces that are not isometric on sections and examples of non-isometric surfaces that are isometric on sections. However, they have non-empty intersections, i.e., some surfaces are both isometric and isometric on sections. In this paper, we prove the positive definiteness of the found invariant.\u0000Further, conditional external curvature is introduced for “basic” sets, open faces, edges, and vertices. It is proved that the conditional curvature of the polyhedral angle considered is monotonicity and positive definiteness. At the end of the article, the problem of the existence and uniqueness of convex polyhedra with given values of conditional curvatures at the vertices is solved.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":"152 7","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41284815","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 $(M,g)$ be a Riemannian manifold and $(TM,tilde{g})$ be its tangent bundle with the $g-$natural metric. In this paper, a family of metallic Riemannian structures $J$ is constructed on $TM,$ found conditions under which these structures are integrable. It is proved that $(TM,tilde{g},J)$ is decomposable if and only if $(M,g)$ is flat.
{"title":"Metallic Riemannian Structures on the Tangent Bundles of Riemannian Manifolds with $g-$Natural Metrics","authors":"","doi":"10.36890/iejg.1145729","DOIUrl":"https://doi.org/10.36890/iejg.1145729","url":null,"abstract":"Let $(M,g)$ be a Riemannian manifold and $(TM,tilde{g})$ be its tangent bundle with the $g-$natural metric. In this paper, a family of metallic Riemannian structures $J$ is constructed on $TM,$ found conditions under which these structures are integrable. It is proved that $(TM,tilde{g},J)$ is decomposable if and only if $(M,g)$ is flat.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48511279","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 define generalized Darboux frame of a a pseudo null curve $alpha$ lying on a lightlike surface in Minkowski space $mathbb{E}_{1}^{3}$. We prove that $alpha$ has two such frames and obtain generalized Darboux frame's equations. We obtain the relations between the curvature functions of $alpha$ with respect to the Darboux frame and generalized Darboux frames. We also find parameter equations of the Darboux vectors of the Frenet, Darboux and generalized Darboux frames and give the necessary and the sufficient conditions for such vectors to have the same directions. Finally, we present related examples.
{"title":"On generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space","authors":"","doi":"10.36890/iejg.1269538","DOIUrl":"https://doi.org/10.36890/iejg.1269538","url":null,"abstract":"In this paper we define generalized Darboux frame of a a pseudo null curve $alpha$ lying on a\u0000lightlike surface in Minkowski space $mathbb{E}_{1}^{3}$. We prove that $alpha$ has two such frames and obtain generalized Darboux frame's equations. We obtain\u0000the relations between the curvature functions of $alpha$ with respect to\u0000 the Darboux frame and generalized Darboux frames. We also find parameter equations of the Darboux vectors of the Frenet, Darboux and generalized Darboux frames and give the necessary and the sufficient conditions for such vectors to have the same directions. Finally, we present related examples.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45018897","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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