TAILIEUCHUNG - B. Y. Chen inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature

In this paper, we prove B. Y. Chen inequalities for submanifolds of a Riemannian manifold of quasiconstant curvature, ., relations between the mean curvature, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered. | Turk J Math 35 (2011) , 501 – 509. ¨ ITAK ˙ c TUB doi: B. Y. Chen inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature ¨ ur Cihan Ozg¨ Abstract In this paper, we prove B. Y. Chen inequalities for submanifolds of a Riemannian manifold of quasiconstant curvature, ., relations between the mean curvature, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered. Key Words: Riemannian manifold of quasi-constant curvature, B. Y. Chen inequality, Ricci curvature 1. Introduction In [11], B. Y. Chen and K. Yano introduced the notion of a Riemannian manifold (M, g) of quasi-constant curvature as a Riemannian manifold with the curvature tensor satisfying the condition R(X, Y, Z, W ) = a [g(Y, Z)g(X, W ) − g(X, Z)g(Y, W )] + +b [g(X, W )T (Y )T (Z) − g(X, Z)T (Y )T (W )+ g(Y, Z)T (X)T (W ) − g(Y, W )T (X)T (Z)] , () where a, b are scalar functions and T is a 1 -form defined by g(X, P ) = T (X), () and P is a unit vector field. It can be easily seen that, if the curvature tensor R is of the form (), then the manifold is conformally flat. If b = 0 then the manifold reduces to a space of constant curvature. A non-flat Riemannian manifold (M n , g) (n > 2) is defined to be a quasi-Einstein manifold [4] if its Ricci tensor satisfies the condition S(X, Y ) = ag(X, Y ) + bA(X)A(Y ), where a, b are scalar functions such that b = 0 and A is a non-zero 1 -form such that g(X, U ) = A(X) for every vector field X and U is a unit vector field. If b = 0 then the manifold reduces to an Einstein manifold. It can be easily seen that every Riemannian manifold of quasi-constant curvature is a quasi-Einstein manifold. 2000 AMS Mathematics Subject Classification: 53C40, 53B05, 53B15. 501 ¨ ¨ OZG UR One of the basic problems in submanifold theory is to find simple relations between the extrinsic and intrinsic invariants of a submanifold. In [6], [7], [9] and

• Toán học
• Riemannian manifold of quasi constant curvature
• B
• Y
• Chen inequality
• Ricci curvature
• Quasiconstant curvature
• Sectional curvatures