Submanifolds with nonparallel first normal bundle revisited

TL;DR

This paper investigates Euclidean submanifolds with nonparallel first normal bundles, showing that slow variation in osculating spaces implies the submanifold is a special ruled type, with estimates on ruling dimensions.

Contribution

It provides a new characterization of submanifolds with nonparallel first normal bundles based on the rate of change of osculating spaces.

Findings

Submanifolds with slowly changing osculating spaces are ruled submanifolds.

A sharp estimate of the dimension of rulings is established.

The geometric structure of such submanifolds is clarified.

Abstract

In this paper, we analyze the geometric structure of an Euclidean submanifold whose osculating spaces form a nonconstant family of proper subspaces of the same dimension. We prove that if the rate of change of the osculating spaces is small, then the submanifold must be a (submanifold of a) ruled submanifold of a very special type. We also give a sharp estimate of the dimension of the rulings.