Special Slant Surfaces and a Basic Inequality

Abstract

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $іdetilde M^2(4∬silon)$ with constant holomorphic sectional curvature 4∊, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H2(p) ≥ 2K(p) − 2(1 + 3 cos2θ)∊. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.