Slant submanifolds of complex projective and complex hyperbolic spaces

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 CP2 and in CH2 and many proper slant minimal surfaces in C-2. In contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces in CP2 and in CH2. 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 CP2 and of CH2. 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 theta -slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.