In this article, we establish an inequality between the sectional curvature function K and the shape operator AH at the mean curvature vector for slant submanifolds in generalized complex space forms. Also a sharp relationship between the k-Ricci curvature and the shape operator AH is proved. | Turk J Math 27 (2003) , 509 – 523. ¨ ITAK ˙ c TUB Shape Operator AH for Slant Submanifolds in Generalized Complex Space Forms Adela Mihai∗ Abstract In this article, we establish an inequality between the sectional curvature function K and the shape operator AH at the mean curvature vector for slant submanifolds in generalized complex space forms. Also a sharp relationship between the k-Ricci curvature and the shape operator AH is proved. Key Words: Shape operator, slant submanifolds, generalized complex space form, k-Ricci curvature. 1. Preliminaries In the introduction of [2], B. Y. Chen recalls as one of the basic problems in submanifold theory: “Find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold”. In the above mentioned paper, B. Y. Chen establishes a relationship between sectional curvature function K and the shape operator AH for submanifolds in real space forms. Also, in [3], B. Y. Chen proves a sharp inequality between the k-Ricci curvature and the shape operator AH . Mathematics Subject Classification 2000 53C40, 53C15. by a JSPS postdoctoral fellowship. ∗ Supported 509 MIHAI In [6], we establish a relationship between the sectional curvature function K and the shape operator AH and a sharp relationship between the k-Ricci curvature and the shape operator AH , respectively, for slant submanifolds in complex space forms. f be an almost Hermitian manifold with almost complex structure J and RieLet M e the operator of covariant differentiation with respect mannian metric g. One denotes by ∇ f. to g in M Definition. If the almost complex structure J satisfies e Y J)X = 0, e X J)Y + (∇ (∇ f, then the manifold M f is called a nearly-Kaehler for any vector fields X and Y on M manifold [5], [11]. Remark. The above condition is equivalent to e X J)X = 0, (∇ f. ∀X ∈ ΓT M f, the Nijenhuis tensor field is For an almost complex structure J on the manifold M defined by NJ (X, Y ) = [JX, JY ] − .
