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Reeb vector field of almost contact metric structure as affine motion 近似接触度量结构的Reeb矢量场为仿射运动
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2022-53-6
L. Ignatochkina
{"title":"Reeb vector field of almost contact metric structure as affine motion","authors":"L. Ignatochkina","doi":"10.5922/0321-4796-2022-53-6","DOIUrl":"https://doi.org/10.5922/0321-4796-2022-53-6","url":null,"abstract":"Smooth manifold with almost contact metric structure (i. e., almost contact metric manifold) was considered in this paper. We used a modern version of Cartan’s method of external forms to conduct our study. We assume that its Reeb vector field is affine motion. We got formulas for components of second covariant differential of contact form for an arbi­trary almost contact metric manifold. Criterion for affine motion of Reeb vector field has been obtained for arbitrary almost contact metric mani­fold in this paper. It is proved that if Reeb vector field of almost contact structure is affine motion then sixth structural tensor of almost contact metric structure is vanishing. It is proved that if Reeb vector field is affine motion and torse-forming vector field then Reeb vector field is Killing vector field. It is proved that if Reeb vector field of almost contact metric structure is torse-forming vector field and it is not Killing vector field then it is not affine motion.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130161055","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}
引用次数: 0
Lie algebras of differentiations of linear algebras over a field 域上线性代数的微分的李代数
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2021-52-12
A. Y. Sultanov, M. Glebova, O.V. Bolotnikova
{"title":"Lie algebras of differentiations of linear algebras over a field","authors":"A. Y. Sultanov, M. Glebova, O.V. Bolotnikova","doi":"10.5922/0321-4796-2021-52-12","DOIUrl":"https://doi.org/10.5922/0321-4796-2021-52-12","url":null,"abstract":"In this paper, we study a system of linear equations that define the Lie algebra of differentiations DerA of an arbitrary finite-dimensional linear algebra over a field. A system of equations is obtained, which is satisfied by the components of an arbitrary differentiation with respect to a fixed basis of algebra A. This system is a system of linear homogeneous equa­tions. The law of transformation of the matrix of this system is proved. The invariance of the rank of the matrix of this system in the transition to a new basis in algebra is proved. Next, we consider the possibility of ap­plying the obtained results in differential geometry when estimating the dimensions of groups of affine transformations from above. As an exam­ple, the method of I. P. Egorov is given for studying the dimensions of Lie algebras of affine vector fields on smooth manifolds equipped with linear connections having non-zero torsion tensor fields.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132893755","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}
引用次数: 1
On the local representation of synectic connections on Weil bundles Weil束上合成连接的局部表示
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2022-53-11
A. Y. Sultanov, G. A. Sultanova
{"title":"On the local representation of synectic connections on Weil bundles","authors":"A. Y. Sultanov, G. A. Sultanova","doi":"10.5922/0321-4796-2022-53-11","DOIUrl":"https://doi.org/10.5922/0321-4796-2022-53-11","url":null,"abstract":"Synectic extensions of complete lifts of linear connections in tangent bundles were introduced by A. P. Shirokov in the seventies of the last century [1; 2]. He established that these connections are linear and are real realizations of linear connections on first-order tangent bundles en­do­wed with a smooth structure over the algebra of dual numbers. He also pro­ved the existence of a smooth structure on tangent bundles of arbitrary or­der on a smooth manifold M over the algebra of plu­ral numbers. Studying holomorphic linear connections on over an algebra , A. P. Shirokov obtained real realizations of these con­nec­tions, which he called Synectic extensions of a linear connection defi­ned on M. A natural generalization of the algebra of plural numbers is the A. Weyl algebra, and a generalization of the tangent bundle is the A. Weyl bundle. It was shown in [3] that a synectic extension of linear connections defined on M a smooth manifold can also be constructed on A. Weyl bundles , where is the A. Weyl algebra. The geometry of these bundles has been studied by many authors — A. Morimoto, V. V. Shu­rygin and others. A detailed analysis of these works can be found in [3]. In this paper, we study synectic lifts of linear connections defined on A. Weyl bundles.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115661017","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}
引用次数: 0
Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection 具有广义Tanaka - Webster连接的非完整Kenmotsu流形
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2020-52-5
A. Bukusheva
{"title":"Non-holonomic Kenmotsu manifolds equipped with generalized Tanaka — Webster connection","authors":"A. Bukusheva","doi":"10.5922/0321-4796-2020-52-5","DOIUrl":"https://doi.org/10.5922/0321-4796-2020-52-5","url":null,"abstract":"А non-holonomic Kenmotsu manifold equipped with a connection analogous to the generalized Tanaka — Webster connection, is consid­ered. The studied connection is obtained from the generalized Tanaka — Webster connection by replacing the first structural endomorphism by the second structural endomorphism. The obtained connection is also called in the work the generalized Tanaka — Webster connection.\u0000\u0000Unlike a Kenmotsu manifold, the structure form of a non-holonomic Kenmotsu manifold is not closed. The consequence of this single differ­ence is a significant discrepancy in the properties of such manifolds. For example, it is proved in the paper that the alternation of the Ricci-Schouten tensor of a non-holonomic Kenmotsu manifold, which is a transverse analogue of the Ricci tensor, is proportional to the external differential of the structural form. At the same time, in the classical case of a Kenmotsu manifold, the Ricci — Schouten tensor is a symmetric tensor.\u0000\u0000It is proved that a Tanaka — Webster connection is a metric connec­tion. It is also proved that from the fact that the alternation of the Ricci-Schouten tensor is proportional to the external differential of the structur­al form, the following statement holds: if a non-holonomic Kenmotsu manifold is an Einstein manifold with respect to the generalized Tanaka — Webster connection, then it is Ricci-flat with respect to the same con­nection.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125972778","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}
引用次数: 2
Fields of fundamental and embracing geometric objects of a regular hyperband with central framing of a projective space 具有射影空间中心框架的规则超带的基本和包含几何对象的场
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2022-53-5
N. A. Eliseeva, Yu. I. Popov
{"title":"Fields of fundamental and embracing geometric objects of a regular hyperband with central framing of a projective space","authors":"N. A. Eliseeva, Yu. I. Popov","doi":"10.5922/0321-4796-2022-53-5","DOIUrl":"https://doi.org/10.5922/0321-4796-2022-53-5","url":null,"abstract":"The study of hyperbands and their generalizations in spaces with dif­ferent fundamental groups is of great interest in connection with numer­ous applications in mathematics and physics. In this paper, we study a special class of hyperbands, i. e., centrally equipped hyperbands. A hy­perband Hm (m ≥ 2) is said to be centrally rigged if the rigging lines in the normals of the 1st kind of the base surface pass through one (the center of the rigging). The article gives a task of a centrally equipped hyperband in the 1st order frame. A sequence of fundamental geometric objects of a hyperstrip with central framing is constructed. An existence theorem for a hyperband with a central framing is proved. It is proved that a hyperstrip with central framing and framing in the sense of Cartan induces a projective connec­tion obtained by projection, where the projection center at each point is the Cartan plane. The spans of the components of the curvature-torsion tensor of the constructed connection are found.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127376180","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}
引用次数: 0
About differential equations of the curvature tensorsof a fundamental group and affine connections 关于基本群的曲率张量微分方程和仿射连接
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2019-50-15
N. Ryazanov
{"title":"About differential equations of the curvature tensors\u0000of a fundamental group and affine connections","authors":"N. Ryazanov","doi":"10.5922/0321-4796-2019-50-15","DOIUrl":"https://doi.org/10.5922/0321-4796-2019-50-15","url":null,"abstract":"The principal bundle is considered, the base of which is an n-dimensional smooth manifold, and the typical fiber is an r-fold Lie group. Structure equations for the forms of the fundamental group and affine connections are given, each of which contains the corresponding components of the curvature tensor. For each connection, an approach is shown that allows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentiating the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solving cubic equations, first by Laptev’s lemma, then by Cartan’s lemma. Taking into account the comparisons modulo basic forms, we obtain already known results (see [3]). Thus, differential equations are derived for the components of the curvature tensor of the first-order fundamentalgroup connection, as well as for the components of the curvature tensor of the affine connection.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127202080","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}
引用次数: 0
To L. V. Stepanova’s anniversary 为l·v·斯捷潘诺娃的周年纪念日干杯
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2019-50-2
G. A. Banaru
{"title":"To L. V. Stepanova’s anniversary","authors":"G. A. Banaru","doi":"10.5922/0321-4796-2019-50-2","DOIUrl":"https://doi.org/10.5922/0321-4796-2019-50-2","url":null,"abstract":"The most important achievements of the outstanding Smolensk geometer\u0000Lidia Vasil’evna Stepanova are presented.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115674449","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}
引用次数: 0
About the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds 二维和三维流形上仿射连接的扭转张量
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2021-52-9
K. Polyakova
{"title":"About the torsion tensor of an affine connection on two-dimensional and three-dimensional manifolds","authors":"K. Polyakova","doi":"10.5922/0321-4796-2021-52-9","DOIUrl":"https://doi.org/10.5922/0321-4796-2021-52-9","url":null,"abstract":"The basis for this study of affine connections in linear frame bundle over a smooth manifold is the structure equations of the bundle. An affine connection is given in this bundle by the Laptev — Lumiste method. The differential equations are written for components of the deformation ten­sor from an affine connection to the symmetrical canonical one. The ex­pressions for the components of the torsion tensor for two-dimensional and three-dimensional manifolds were found.\u0000\u0000For a two-dimensional manifold, the affine torsion is a fraction, in the numerator there is a linear combination of two fiber coordinates which coefficients are two functions depending on the base coordinates (the co­ordinates on the base), and in the denominator there is the determinant composed of the fiber coordinates (the coordinates in a fiber). For a three-dimensional manifold, the arbitrariness of the numerator is determined by nine functions depending on the base coordinates.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125932839","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}
引用次数: 0
On some extension of the second order tangent space for a smooth manifold 光滑流形的二阶切空间的扩展
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2022-53-9
K. Polyakova
{"title":"On some extension of the second order tangent space for a smooth manifold","authors":"K. Polyakova","doi":"10.5922/0321-4796-2022-53-9","DOIUrl":"https://doi.org/10.5922/0321-4796-2022-53-9","url":null,"abstract":"This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold. Using the perturbation of the exterior derivative and ordinary diffe­ren­tial, mappings are introduced that enable us to construct non-sym­met­rical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimen­sional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold. A deformed external differential is widely used, which is a differen­tial, i. e., its reapplication vanishes. We introduce a deformed external dif­ferential being a differential along the curves on the manifold, i. e., its re­peated application along the curves on the manifold gives zero.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129394245","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}
引用次数: 1
On geometry of orbits of adapted projective framespace 自适应射影框架空间的轨道几何
Differential Geometry of Manifolds of Figures Pub Date : 1900-01-01 DOI: 10.5922/0321-4796-2019-50-11
A. Kuleshov
{"title":"On geometry of orbits of adapted projective frame\u0000space","authors":"A. Kuleshov","doi":"10.5922/0321-4796-2019-50-11","DOIUrl":"https://doi.org/10.5922/0321-4796-2019-50-11","url":null,"abstract":"The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The ndimensional projective space with a distinguished point (the center) is considered. The action of matrix affine group of order n on the adapted projective frame manifold is given. It is shown that the linear frames, i. e., bases of the tangent space, can be identified with the orbits of adapted projective frames under the action of some normal subgroup of this group. Two adapted frames are said to be equivalent if they belong to the same orbit. The strict perspectivity relation between two adapted frames is introduced. The proofs of the theorem on the Desargues hyperplane and of the criterion of equivalence are simplified. According to this criterion, two adapted frames in strict perspective are equivalent if and only if the Desargues hyperplane generated by these frames is passing through the center.","PeriodicalId":114406,"journal":{"name":"Differential Geometry of Manifolds of Figures","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126079228","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}
引用次数: 0
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