Sectional curvature of contact $CR$-submanifolds of an odd-dimensional unit sphere

Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 777-787 Printed December 1, 2005

Hyang Sook Kim and Jin Suk Pak Inje University, Kyungpook National University

Abstract : 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