Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

Abstract

The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then, there have been rapid developments in the theory of finite type. The simplest finite type submanifolds and submanifolds with finite type Gauss maps are those which are of 1-type. The classes of such submanifolds constitute very special and interesting families in the finite type theory. The purpose of this paper is thus to provide a comprehensive survey on 1-type submanifolds and submanifolds with 1-type Gauss maps done during the last forty years.