Nonnegatively curved Euclidean submanifolds in codimension two

Abstract

We provide a classification of compact Euclidean submanifolds $M^n\subset{\mathbb{R}}^{n+2}$ with nonnegative sectional curvature, for $n\ge 3$. The classification is in terms of the induced metric (including the diffeomorphism classification of the manifold), and we study the structure of the immersions as well. In particular, we provide the first known example of a nonorientable quotient $({\mathbb{S}}^{n-1}\times{\mathbb{S}}^1)/{{\mathbb{Z}}_2}\subset{\mathbb{R}}^{n+2}$ with nonnegative curvature. For the 3-dimensional case, we show that either the universal cover is isometric to ${\mathbb{S}}^2\times{\mathbb{R}}$, or $M^3$ is diffeomorphic to a lens space, and the complement of the (nonempty) set of flat points is isometric to a twisted cylinder $(N^2\times{\mathbb{R}})/{\mathbb{Z}}$. As a consequence we conclude that, if the set of flat points is not too big, there exists a unique flat totally geodesic surface in $M^3$ whose complement is the union of one or two twisted cylinders over disks.