Slant submanifolds ofcomplex projective and complex hyperbolic spaces

Abstract

An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In [1–4,7,8] slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, 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 general construction procedure for obtaining the explicit expressions of such slant submanifolds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP^2 and of CH^2. Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special θ-slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.