Explicit immersions of surfaces in R-4 with arbitrary constant Jordan angles

摘要

An immersed surface in R-4 is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in R-4 has been proved to exist, Bayard et al. (Geom Dedicata 162:153-176, 2013), but there are only explicit examples of non planar surfaces for the extremal angles 0 and pi/2 as the Clifford torus. In this work, an alternative proof of the existence has been obtained that is based on the solution of a hyperbolic partial differential equation. Finally, after a study of the known solutions of the hyperbolic equation, an explicit expression with arbitrary CJA is provided for a family of immersions with an additional geometric property written in terms of the local invariants.