Bulletin of the
Korean Mathematical Society BKMS

In this paper we study $(n+1)$-dimensional compact contact $CR$-submanifolds of $(n-1)$ contact $CR$-dimension immersed in an odd-dimensional unit sphere $S^{2m+1}$. Especially we provide necessary conditions in order for such a submanifold to be the generalized Clifford surface $$ S^{2n_1 +1}(((2n_1 +1)/(n+1))^{\frac{1}{2}})\times S^{2n_2 +1}(((2n_2 +1)/(n+1))^{\frac{1}{2}}) $$ for some portion $(n_1 ,n_2)$ of $(n-1)/2$ in terms with sectional curvature.

Keywords: Sasakian manifold, odd-dimensional unit sphere, contact $CR$-submanifold, contact $CR$-dimension, minimal real hypersurface, sectional curvature

MSC numbers: 53C40, 53C15