2011/05/09 by Bayard, Pierre, Di Scala, Antonio J., Osuna-Castro, Osvaldo +1
#53C40 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1105.1791
We study surfaces in \R4 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 \R2. 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 \R4 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.