Pythagorean Submanifolds in Euclidean Spaces

Abstract

In this paper we introduce the notion of a Pythagorean sub manifold isometrically immersed into a Euclidean space. This definiton will be based on the shape operator of the submanifold. We prove that any Pythagorean submanifold in a Euclidean space is isoparametric, the principal curvatures along any parallel normal vector field being given in terms of the golden or conjugate golden ratios. In addition, we obtain that a Pythagorean submanifold of codimension 1 (resp. = 2) is totally umbilical (resp. pseudo-umbilical), classifying those submanifolds.