Surfaces in $mathbb{R}^4$ with constant principal angles with respect to a plane

TL;DR

This paper investigates surfaces in four-dimensional space with constant principal angles relative to a plane, establishing existence results, classifications, and characterizations of such surfaces, including special cases like spheres and parallel mean curvature vectors.

Contribution

It proves the existence of surfaces with arbitrary constant principal angles via PDEs, classifies surfaces with specific angle conditions, and links their existence to symplectomorphisms in .

Findings

Surfaces with one principal angle zero are unions of normal holonomy tubes.

Complete constant angle surfaces are extrinsic products.

Surfaces with constant principal angles in a sphere or with parallel mean curvature are classified.

Abstract

We study surfaces in $\R^4$ whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We classify all surfaces with one principal angle equal to 0 and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in $\R^4$ with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector field.