No skew branes on non-degenerate hyperquadrics

TL;DR

This paper proves that non-degenerate hyperquadrics in Euclidean space cannot have skew branes, meaning any compact immersed hypersurface must have parallel tangent spaces at two distinct points, without assuming smoothness or genericity.

Contribution

It establishes the non-existence of skew branes on non-degenerate hyperquadrics without smoothness or genericity constraints, extending previous results.

Findings

Non-degenerate hyperquadrics admit no skew branes.

Any compact immersed hypersurface on such hyperquadrics has parallel tangent spaces at two points.

Results hold without smoothness or genericity assumptions.

Abstract

We show that non-degenerate hyperquadrics in R^{n+2} admit no skew branes. Stated more traditionally, a compact codimension-one immersed submanifold of a non-degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points. Similar results have been proven by others, but (except for ellipsoids in R^3) always under C^2 smoothness and genericity assumptions. We use neither assumption here.