Slant submanifolds ofcomplex projective and complex hyperbolic spaces

Abstract

An interestingclass of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifoldswith constant Wirtinger angle. In [1–4,7,8] slantsubmanifolds of complex projective and complex hyperbolic spaces have been investigated. Inparticular, it was shown that there exist many proper slant surfaces in CP^2 and inCH^2 and many proper slant minimal surfaces in C². In contrast,in the first part of this paper we prove that there do not exist proper slant minimal surfaces inCP^2 and in CH^2. In the second part, we present a generalconstruction procedure for obtaining the explicit expressions of such slant submanifolds. By applyingthis general construction procedure, we determine the explicit expressions of special slant surfacesof CP^2 and of CH^2. Consequently, we are able to completelydetermine the slant surface which satisfies a basic equality. Finally, we apply the constructionprocedure to prove that special θ-slant isometric immersions of a hyperbolicplane into a complex hyperbolic plane are not unique in general.