Submanifolds of higher rank with curvature normals of constant length

Abstract

We study the local geometry of Euclidean submanifolds of rank at least two whose curvature normals with respect to the flat part of the normal bundle have constant length, equivalently, whose adapted third fundamental form has constant eigenvalues. The complete case was fully settled by Di Scala and the third author, who showed that such a complete submanifold has constant principal curvatures. Around a regular point, we show that $M$ is generated, via parallel manifolds, by a hypersurface $N$ whose extrinsic factors are contained in spheres and, when of dimension at least two, have rank one; conversely, such factors can be assembled to construct examples of $M$. The proof relies on the isoparametric rank theorem to rule out non-umbilical factors. As a further application, when the inner products of the curvature normals are constant, we show that there are exactly two eigendistributions, one of which is one-dimensional and autoparallel, extending, for arbitrary codimension when the normal bundle is flat, curvature-homogeneity results of Tsukada and the recent ones of Bryant, Florit, and Ziller for hypersurfaces.