A Characterization of Homogeneous Holomorphic Two-Spheres in $Q_n$

Abstract

In this paper, we classify holomorphic curves in $Q_n$ with certain geometric conditions. We study the (1,0) part of $k{\text {th}}$ covariant derivative about the second fundamental form denoted by $\mathbf{a }_{,k}$, $0\le k\le [\frac{n}{2}]-2$; the norm of its symmetric product is denoted by $\tau _k=|\mathbf{a }_{,k} \cdot \mathbf{a }_{,k}|$. It is proven that a holomorphic curve in $Q_n$ is homogeneous if the Gaussian curvature, the norm of the second fundamental form and $\tau _k$ are all constant. Moreover, all the homogeneous holomorphic curves are uniquely determined by our given examples, up to a rigid motion of $Q_n$.