光滑流形的二阶切空间的扩展

K. Polyakova
{"title":"光滑流形的二阶切空间的扩展","authors":"K. Polyakova","doi":"10.5922/0321-4796-2022-53-9","DOIUrl":null,"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":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"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\":null,\"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\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry of Manifolds of Figures\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5922/0321-4796-2022-53-9\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry of Manifolds of Figures","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5922/0321-4796-2022-53-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1

摘要

本文涉及微分几何,研究方法基于拉普捷夫(g.f. Laptev)的扩展包络法,该方法推广了卡坦(e.c artan)的运动框架和外部形式方法。我们考虑一个光滑的m维流形,它的正切和余切空间,以及这个流形上的二阶坐标系和二阶协坐标系。利用外导数和常微分的摄动,引入映射,使我们能够在光滑流形上构造非对称二阶坐标系和协坐标系。证明了二阶切空间向光滑m维流形的扩展是通过将流形上线性框架束的垂直向量与该流形的二阶切向量相加来实现的。一种变形的外差速器被广泛使用,它是一个差速器,也就是说,它的重复应用消失了。我们引入了一种变形的外微分,它是沿流形上曲线的微分,即它沿流形上曲线的重复应用得到零。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On some extension of the second order tangent space for a smooth manifold
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.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信