{"title":"Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping","authors":"O. O. Shugailo","doi":"10.30970/ms.60.1.99-112","DOIUrl":null,"url":null,"abstract":"In the paper we study equiaffine immersions $f\\colon (M^n,\\nabla) \\rightarrow {\\mathbb{R}}^{n+2}$ with flat connection $\\nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $f\\colon ({M}^n,\\nabla)\\rightarrow({\\mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $\\nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:$(i)$ $\\vec{r}=g(u^1,\\ldots,u^n) \\vec{a}_1+\\int\\vec{\\varphi}(u^1)du^1+\\sum\\limits_{i=2}^n u^i\\vec{a}_i;$$(ii)$ $\\vec{r}=(g(u^2,\\ldots,u^n)+u^1)\\vec{a}+\\int v(u^1) \\vec{\\eta}(u^1)du^1+\\sum\\limits_{i=2}^n u^i\\int\\lambda_i(u^1)\\vec{\\eta}(u^1)du^1;$$(iii)$ $\\vec{r}=(g(u^2,\\ldots,u^n)+u^1)\\vec{\\rho}(u^1)+\\int (v(u^1) - u^1)\\dfrac{d \\vec{\\rho}(u^1)}{d u^1}du^1+\\sum\\limits_{i=2}^n u^i\\int\\lambda_i(u^1)\\dfrac{d \\vec{\\rho}(u^1)}{d u^1}du^1.$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.60.1.99-112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
In the paper we study equiaffine immersions $f\colon (M^n,\nabla) \rightarrow {\mathbb{R}}^{n+2}$ with flat connection $\nabla$ and one-dimensional Weingarten mapping. For such immersions there are two types of the transversal distribution equiaffine frame.We give a parametrization of a submanifold with the given properties for both types of equiaffine frame. The main result of the paper is contained in Theorems 1, 2 and Corollary 1: Let $f\colon ({M}^n,\nabla)\rightarrow({\mathbb{R}}^{n+2},D)$ be an affine immersion with pointwise codimension 2, equiaffine structure, flat connection $\nabla$, one-dimensional Weingarten mapping then there exists three types of its parametrization:$(i)$ $\vec{r}=g(u^1,\ldots,u^n) \vec{a}_1+\int\vec{\varphi}(u^1)du^1+\sum\limits_{i=2}^n u^i\vec{a}_i;$$(ii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{a}+\int v(u^1) \vec{\eta}(u^1)du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\vec{\eta}(u^1)du^1;$$(iii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{\rho}(u^1)+\int (v(u^1) - u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1.$