In this article, we establish inequalities between the Ricci curvature and the squared mean curvature, and also between the k-Ricci curvature and the scalar curvature for a slant, semi-slant and bi-slant submanifold in a cosymplectic space form of constant ϕ- sectional curvature with arbitrary codimension. | Turk J Math 30 (2006) , 43 – 56. ¨ ITAK ˙ c TUB Inequality for Ricci Curvature of Slant Submanifolds in Cosymplectic Space Forms Dae Won Yoon Abstract In this article, we establish inequalities between the Ricci curvature and the squared mean curvature, and also between the k-Ricci curvature and the scalar curvature for a slant, semi-slant and bi-slant submanifold in a cosymplectic space form of constant ϕ- sectional curvature with arbitrary codimension. Key Words: Mean curvature, sectional curvature, k-Ricci curvature, slant submanifold, semi-slant submanifold, bi-slant submanifold, cosymplectic space form. 1. Introduction ˜ be a (2m + 1)-dimensional almost contact manifold endowed with an almost Let M contact structure (ϕ, ξ, η), that is, ϕ is a (1,1) tensor field, ξ is a vector field and η is a 1-form such that ϕ2 = −I + η ⊗ ξ and η(ξ) = 1. Then, ϕ(ξ) = 0 and η ◦ ϕ = 0. Let g be a compatible Riemannian metric with (ϕ, ξ, η), that is, g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ) or equivalent, g(X, ϕY ) = −g(ϕX, Y ) and ˜ . Then, M ˜ becomes an almost contact metric manifold g(X, ξ) = η(X) for all X, Y ∈ M equipped with an almost contact metric structure (ϕ, ξ, η, g). An almost contact metric ˜ is the Levi-Civita connection of the ˜ X ϕ = 0, where ∇ manifold is cosymplectic ([1]) if ∇ ˜ X ξ = 0. ˜ X ϕ = 0 it follows that ∇ Riemannian metric g. From the formula ∇ 2000 AMS Mathematics Subject Classification: 53B25, 53D10. 43 YOON ˜ of an almost contact metric manifold M ˜ is called a ϕ-section A plane section π in Tp M ˜ is of constant ϕ-sectional curvature if sectional curvature K(π) ˜ if π ⊥ ξ and ϕ(π) = π. M ˜ and the choice of a point does not depend on the choice of the ϕ-section π of Tp M ˜ . A cosymplectic manifold M ˜ is said to be a cosymplectic space form if the ϕp ∈ M ˜ A cosymplectic space form will be denoted by sectional curvature is constant c along M. ˜ (c). Then the Riemannian curvature tensor R ˜ on M(c) ˜ M is given by ([9]) ˜ 4R(X, Y, Z, W ) .
