{"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}
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.