Equiaffine immersions of codimension two with flat connection and one-dimensional Weingarten mapping

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)$ $\vec{r}=(g(u^2,\ldots,u^n)+u^1)\vec{\rho}(u^1)+\int (v(u^1) - u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1+\sum\limits_{i=2}^n u^i\int\lambda_i(u^1)\dfrac{d \vec{\rho}(u^1)}{d u^1}du^1.$