Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping
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...
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Ivan Franko National University of Lviv
2023-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/297 |
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| author | O. O. Shugailo |
| author_facet | O. O. Shugailo |
| author_sort | O. O. Shugailo |
| collection | DOAJ |
| description | 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:
$(i)$
$\vec{r}=g(u^1,\ldots,u^n) \vec{a}_1+\int
\vec{\varphi}(u^1)du^1+\sum\limits_{i=2}^n u^i\vec{a}_i;$
$(ii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{a}+\int v(u^1) \vec{\eta}(u^1)du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\vec{\eta}(u^1)du^1;$
$(iii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{\rho}(u^1)+\int (v(u^1) - u^1)
\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1.$ |
| format | Article |
| id | doaj-art-917e7690a77046098c466c625fae89e1 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-09-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-917e7690a77046098c466c625fae89e12025-08-20T02:41:29ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-09-016019911210.30970/ms.60.1.99-112297Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mappingO. O. Shugailo0School of Mathematics and Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, UkraineIn 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: $(i)$ $\vec{r}=g(u^1,\ldots,u^n) \vec{a}_1+\int \vec{\varphi}(u^1)du^1+\sum\limits_{i=2}^n u^i\vec{a}_i;$ $(ii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{a}+\int v(u^1) \vec{\eta}(u^1)du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\vec{\eta}(u^1)du^1;$ $(iii)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{\rho}(u^1)+\int (v(u^1) - u^1) \dfrac{d \vec{\rho}(u^1)}{d u^1}du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1.$http://matstud.org.ua/ojs/index.php/matstud/article/view/297affine immersion; flat connection; equiaffine structure; weingarten mapping |
| spellingShingle | O. O. Shugailo Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping Математичні Студії affine immersion; flat connection; equiaffine structure; weingarten mapping |
| title | Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping |
| title_full | Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping |
| title_fullStr | Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping |
| title_full_unstemmed | Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping |
| title_short | Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping |
| title_sort | equiaffine immersions of codimension two with flat connection and one dimensional weingarten mapping |
| topic | affine immersion; flat connection; equiaffine structure; weingarten mapping |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/297 |
| work_keys_str_mv | AT ooshugailo equiaffineimmersionsofcodimensiontwowithflatconnectionandonedimensionalweingartenmapping |