The Lie group over the hyper-dual matrices and its corresponding Lie algebra are first introduced in this study. One of Euler's strategies called the Euler-Rodrigues formula is applied to the matrices of hyper-dual rotations. The fundamental relationship between the hyper-dual numbers and the dual numbers allows us to apply the formula on dual lines and two intersecting real lines in the three-dimensional dual and Euclidean spaces, respectively.
{"title":"The Application of Euler-Rodrigues Formula Over Hyper-Dual Matrices","authors":"Çağla Ramis, Y. Yaylı, İrem Zengi̇n","doi":"10.36890/iejg.1127216","DOIUrl":"https://doi.org/10.36890/iejg.1127216","url":null,"abstract":"The Lie group over the hyper-dual matrices and its corresponding Lie algebra are first introduced in this study. One of Euler's strategies called the Euler-Rodrigues formula is applied to the matrices of hyper-dual rotations. The fundamental relationship between the hyper-dual numbers and the dual numbers allows us to apply the formula on dual lines and two intersecting real lines in the three-dimensional dual and Euclidean spaces, respectively.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42883996","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 introduce the Cheeger-Gromoll type metric on the coframe bundle of a Riemannian manifold and investigate the Levi-Civita connection, curvature tensor, sectional curvature and geodesics of coframe bundle with this metric.
{"title":"On the Differential Geometry of Coframe Bundle with Cheeger-Gromoll Metric","authors":"Habil Fattayev","doi":"10.36890/iejg.1071782","DOIUrl":"https://doi.org/10.36890/iejg.1071782","url":null,"abstract":"In this paper we introduce the Cheeger-Gromoll type metric on the coframe bundle of a\u0000Riemannian manifold and investigate the Levi-Civita connection, curvature tensor, sectional\u0000curvature and geodesics of coframe bundle with this metric.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44652629","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 first define the notion of Lagrangian statistical submersion from a K"ahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from K"ahler-like statistical manifolds onto statistical manifolds and discuss the equality case of the obtained inequality through a basictensor introduced by O'Neill that plays the role of the second fundamental form of an isometric immersion. At the end, we give a nontrivial example of a K"ahler-like statistical submersion.
{"title":"B.-Y. Chen's Inequality for K\"ahler-like Statistical Submersions","authors":"A. Siddiqui","doi":"10.36890/iejg.1006287","DOIUrl":"https://doi.org/10.36890/iejg.1006287","url":null,"abstract":"In this paper, we first define the notion of Lagrangian statistical submersion from a K\"ahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from K\"ahler-like statistical manifolds onto statistical manifolds and discuss the equality case of the obtained inequality through a basictensor introduced by O'Neill that plays the role of the second fundamental form of an isometric immersion. At the end, we give a nontrivial example of a K\"ahler-like statistical submersion.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47287083","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}
{"title":"The de Rham Cohomology Group of a Hemi-Slant Submanifold in Metallic Riemannian Manifolds","authors":"Mustafa Gök","doi":"10.36890/iejg.1118628","DOIUrl":"https://doi.org/10.36890/iejg.1118628","url":null,"abstract":"<jats:p xml:lang=\"tr\" />","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49170207","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 aim of the present paper is to study half-lightlike submanifolds of a semi-Riemannian manifold endowed with a metallic structure. We introduce a special half-lightlike submanifold called screen semi-invariant half lightlike submanifold in metallic semi-Riemannian manifolds and give an example. We present necessary and sufficient conditions for the distributions included in the definition of such half lightlike submanifolds to be integrable. Moreover, we analyze geometry of a screen semi-invariant half lightlike submanifold in a locally metallic semi- Riemannian manifold when it is totally geodesic and screen conformal.
{"title":"Half-Lightlike Submanifolds of Metallic semi-Riemannian Manifolds","authors":"B. E. Acet, F. Erdoğan, Selcen Yüksel Perktaş","doi":"10.36890/iejg.1085596","DOIUrl":"https://doi.org/10.36890/iejg.1085596","url":null,"abstract":"The aim of the present paper is to study half-lightlike submanifolds of a semi-Riemannian\u0000manifold endowed with a metallic structure. We introduce a special half-lightlike submanifold\u0000called screen semi-invariant half lightlike submanifold in metallic semi-Riemannian manifolds\u0000and give an example. We present necessary and sufficient conditions for the distributions\u0000included in the definition of such half lightlike submanifolds to be integrable. Moreover, we\u0000analyze geometry of a screen semi-invariant half lightlike submanifold in a locally metallic semi-\u0000Riemannian manifold when it is totally geodesic and screen conformal.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46133981","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 study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $gamma$ with the tangents of the curves. We examine the relations between torsions and curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=cos (theta/2)-mathbf{i}sin (theta/2)cos gamma -mathbf{j}sin (theta/2)sin gamma$, where $theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.
{"title":"A new generalization of some curve pairs","authors":"Oğuzhan Çeli̇k, M. Ozdemir","doi":"10.36890/iejg.1110327","DOIUrl":"https://doi.org/10.36890/iejg.1110327","url":null,"abstract":"In this study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $gamma$ with the tangents of the curves. We examine the relations between torsions and\u0000curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=cos (theta/2)-mathbf{i}sin (theta/2)cos gamma -mathbf{j}sin (theta/2)sin gamma$, where $theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41887842","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}
{"title":"On a 2-form Derived by Riemannian Metric in the Tangent Bundle","authors":"N. Gurbanova","doi":"10.36890/iejg.1137820","DOIUrl":"https://doi.org/10.36890/iejg.1137820","url":null,"abstract":"<jats:p xml:lang=\"tr\" />","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43531956","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 show that convex hull of extreme points of a closed convex subset of a compact flat Riemannian manifold is equal to the subset itself.
我们证明了紧致平坦黎曼流形的闭凸子集的极值点的凸包等于子集本身。
{"title":"Convex hull of extreme points in flat Riemannian manifolds","authors":"R. Mirzaie","doi":"10.36890/iejg.1046707","DOIUrl":"https://doi.org/10.36890/iejg.1046707","url":null,"abstract":"We show that convex hull of extreme points of a closed convex subset of a compact \u0000flat Riemannian manifold is equal to the subset itself.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46519754","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 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a tile of more complex shape is true.
{"title":"Keller's Conjecture Revisited","authors":"P. Horák, Dongryul Kim","doi":"10.36890/iejg.984269","DOIUrl":"https://doi.org/10.36890/iejg.984269","url":null,"abstract":"In 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a \u0000tile of more complex shape is true.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45968362","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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