Slant Submanifolds in Complex Euclidean Spaces

Abstract

An immersion of a differentiable manifold into an almost Hermitian manifold is called a \textit{general slant immersion} if it has constant Wirtinger angle ([3, 6]). A general slant immersion which is neither holomorphic nor totally real is called a proper slant immersion. In the first part of this article, we prove that every general slant immersion of a compact manifold into the complex Euclidean $m$-space $\mathbf{C}^m$ is totally real. This result generalizes the well-known fact that there exist no compact holomorphic submanifolds in any complex Euclidean space. In the second part, we classify proper slant surfaces in $\mathbf{C}^2$ when they are contained in a hypersphere $S^3$, or contained in a hyperplane $E^3$, or when their Gauss maps have rank $<2$.