Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.
{"title":"On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric","authors":"A. Zagane","doi":"10.36890/iejg.1182395","DOIUrl":"https://doi.org/10.36890/iejg.1182395","url":null,"abstract":"Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.","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":"47611871","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, it's introduced the curves lying on parallel-like surface M^{f} of a surface M in Euclidean space. Taking into account the definition of the parallel-like surface it's obtained parametric expression of these curves and examined the Darboux frame for these curves which we call image curves. And finally, the curves lying on the surfaces M and M^{f} are compared by considering their geodesic and normal curvatures, the geodesic torsion.
{"title":"Image Curves on the Parallel-like Surfaces in E³","authors":"Semra Yurttançikmaz, Ö. Tarakci","doi":"10.36890/iejg.1178434","DOIUrl":"https://doi.org/10.36890/iejg.1178434","url":null,"abstract":"In this paper, it's introduced the curves lying on parallel-like surface M^{f} of a surface M in Euclidean space. Taking into account the definition of the parallel-like surface it's obtained parametric expression of these curves and examined the Darboux frame for these curves which we call image curves. And finally, the curves lying on the surfaces M and M^{f} are compared by considering their geodesic and normal curvatures, the geodesic torsion.","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":"49603867","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 firstly, operators were applied to vertical and horizontal lifts with respect to the diagonal lift ϕ^{D} of tensor fields of type (1,1) from manifold to its tensor bundle of type (p,q) along the cross-section, respectively. Secondly, we get the conditions of almost holomorfic vector field with respect to ϕ^{D} on T_{q}^{p}(M).
{"title":"Operators Applied to Lifts with Respect to the Diagonal Lifts of Affinor Fields Along a Cross-Section on $T_{q}^{p}(M)$","authors":"Haşim Çayır, Behboudi Asl","doi":"10.36890/iejg.1170443","DOIUrl":"https://doi.org/10.36890/iejg.1170443","url":null,"abstract":"In this paper firstly, operators were applied to vertical and horizontal lifts with respect to the diagonal lift ϕ^{D} of tensor fields of type (1,1) from manifold to its tensor bundle of type (p,q) along the cross-section, respectively. Secondly, we get the conditions of almost holomorfic vector field with respect to ϕ^{D} on T_{q}^{p}(M).","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":"41984008","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, the spinor formulations of Mannheim curve pair are investigated. First of all, two spinors matching to Mannheim curve pair are given and by considering the relationships between the Frenet frames of Mannheim curve pair, the relationship between two spinors matching to this curve pair are gotten. Therefore, a geometric interpretations of spinors are obtained using the Mannheim curve pair and considering Mannheim curve as helix the spinor formulations of Mannheim curve pair are given. Moreover, the spinor formulations are also obtained for the curvatures of the Mannheim curve pair. Consequently, an example of these spinors is obtained. Therefore, it is thought that this study will make an important contribution to the mathematical analysis and geometric interpretation of spinors, which have many uses in physics.
{"title":"The Spinor Expressions of Mannheim Curves in Euclidean 3-Space","authors":"Tülay Erişir, Zeynep İsabeyoğlu","doi":"10.36890/iejg.1210442","DOIUrl":"https://doi.org/10.36890/iejg.1210442","url":null,"abstract":"In this paper, the spinor formulations of Mannheim curve pair are investigated. First of all, two spinors matching to Mannheim curve pair are given and by considering the relationships between the Frenet frames of Mannheim curve pair, the relationship between two spinors matching to this curve pair are gotten. Therefore, a geometric interpretations of spinors are obtained using the Mannheim curve pair and considering Mannheim curve as helix the spinor formulations of Mannheim curve pair are given. Moreover, the spinor formulations are also obtained for the curvatures of the Mannheim curve pair. Consequently, an example of these spinors is obtained. Therefore, it is thought that this study will make an important contribution to the mathematical analysis and geometric interpretation of spinors, which have many uses in physics.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44866569","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 geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications in the theory of control systems, in dynamic systems, in geometry and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.
{"title":"The Geometry of Vector Fields and Two Dimensional Heat Equation","authors":"Narmanov ABDUGAPPAR YAKUBOVİCH, Rajabov Eldor","doi":"10.36890/iejg.1230873","DOIUrl":"https://doi.org/10.36890/iejg.1230873","url":null,"abstract":"The geometry of orbits of families of smooth vector fields was studied by many mathematicians due\u0000to its importance in applications in the theory of control systems, in dynamic systems, in geometry\u0000and in the theory of foliations.\u0000In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate\u0000singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion.\u0000\u0000In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47051354","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 paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.
{"title":"Back to Almost Ricci Solitons","authors":"V. Rovenski, S. Stepanov, I. Tsyganok","doi":"10.36890/iejg.1223973","DOIUrl":"https://doi.org/10.36890/iejg.1223973","url":null,"abstract":"In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons.\u0000Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41786509","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 main goal of this study is to bring together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new type frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space $mathbb{E}^3$, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.
{"title":"Spinor Representations of Positional Adapted Frame in the Euclidean 3-Space","authors":"Zehra İşbilir, Kahraman Esen Özen, M. Güner","doi":"10.36890/iejg.1179503","DOIUrl":"https://doi.org/10.36890/iejg.1179503","url":null,"abstract":"The main goal of this study is to bring together the spinors, which have a major place in several disciplines from mathematics to physics, and Positional Adapted Frame (PAF) which is a new type frame that attracts the attention of many researchers. In accordance with this purpose, we introduce the spinor representations for the trajectories endowed with PAF in the Euclidean 3-space $mathbb{E}^3$, and construct the spinor equations of PAF vectors. Then, we find the relations between spinor representations of PAF and Serret-Frenet frame. Also we give some results and present some geometric interpretations with respect to this relationship. Moreover, we present an illustrative numerical example in order to support the given theorems and results.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41730301","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 have studied the curvature properties of the Schouten-van Kampen connection on the n-dimensional Para Sasakian manifold and obtained some new results. Also, we studied projective curvature tensor, concircular curvature tensor, and Nijenhuis tensor for the Para-Sasakian manifold with respect to the Schouten-van Kampen connection.
{"title":"On Para-Sasakian manifold with respect to the Schouten-van Kampen connection","authors":"Shivani Sundri̇yal, J. Upreti̇","doi":"10.36890/iejg.1200729","DOIUrl":"https://doi.org/10.36890/iejg.1200729","url":null,"abstract":"In the present paper, we have studied the curvature properties of the Schouten-van Kampen \u0000connection on the n-dimensional Para Sasakian manifold and obtained some new results. Also, \u0000we studied projective curvature tensor, concircular curvature tensor, and Nijenhuis tensor for the \u0000Para-Sasakian manifold with respect to the Schouten-van Kampen connection.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41790752","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}
Bang‐Yen Chen, Erhan Güler, Y. Yaylı, H. H. Hacisalihoglu
The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then, there have been rapid developments in the theory of finite type. The simplest finite type submanifolds and submanifolds with finite type Gauss maps are those which are of 1-type. The classes of such submanifolds constitute very special and interesting families in the finite type theory.
{"title":"Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map","authors":"Bang‐Yen Chen, Erhan Güler, Y. Yaylı, H. H. Hacisalihoglu","doi":"10.36890/iejg.1216024","DOIUrl":"https://doi.org/10.36890/iejg.1216024","url":null,"abstract":"The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then, there have been rapid developments in the theory of finite type. \u0000The simplest finite type submanifolds and submanifolds with finite type Gauss maps are those which are of 1-type. The classes of such submanifolds constitute very special and interesting families in the finite type theory.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44867321","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 main objective of this paper is to describe the perfect fluid spacetimes fulfilling $f(R)$-gravity, when Ricci-Yamabe, gradient Ricci-Yamabe and $eta$-Ricci-Yamabe solitons are its metrics. We acquire conditions for which the Ricci-Yamabe and the gradient Ricci-Yamabe solitons are expanding, steady or shrinking. Furthermore, we investigate $eta$-Ricci-Yamabe solitons and deduce a Poisson equation and with the help of this equation, we acquire some significant results.
{"title":"Ricci-Yamabe solitons in $f(R)$-gravity","authors":"K. De, U. De","doi":"10.36890/iejg.1234057","DOIUrl":"https://doi.org/10.36890/iejg.1234057","url":null,"abstract":"The main objective of this paper is to describe the perfect fluid spacetimes fulfilling $f(R)$-gravity, when Ricci-Yamabe, gradient Ricci-Yamabe and $eta$-Ricci-Yamabe solitons are its metrics. We acquire conditions for which the Ricci-Yamabe and the gradient Ricci-Yamabe solitons are expanding, steady or shrinking. Furthermore, we investigate $eta$-Ricci-Yamabe solitons and deduce a Poisson equation and with the help of this equation, we acquire some significant results.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44676335","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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