Abstract
In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products.
Highlights
IntroductionA differentiable manifold Mendowed with an almost contact metric structure (φ, ξ, η, g) is said to be nearly cosymplectic if the covariant derivative of the almost contact structure φ with respectis skew-symmetric, i.e.,
A differentiable manifold Mendowed with an almost contact metric structure (φ, ξ, η, g) is said to be nearly cosymplectic if the covariant derivative of the almost contact structure φ with respectis skew-symmetric, i.e., (∇̃ X φ) X = 0, for every vector field X on M.to the Levi-Civita connection ∇These manifolds were defined on the line of nearly Kaehler manifolds and studied by Blair [1], Blair and Showers [2]
We study CR-slant warped product submanifolds of nearly cosymplectic manifolds which are the more general classes of contact metric manifolds
Summary
A differentiable manifold Mendowed with an almost contact metric structure (φ, ξ, η, g) is said to be nearly cosymplectic if the covariant derivative of the almost contact structure φ with respectis skew-symmetric, i.e., Cappelletti-Montano and Dileo [5] proved that every nearly Sasakian manifold of dimension 5 has an associated nearly cosymplectic structure, thereby showing the close relation between these two notions. In 1969, Bishop and O’Neill introduced the notion of a warped product manifolds to provide a class of complete Riemannian manifolds with negative curvature [7]. This scheme was later applied to semi-Riemannian geometry and the theory of relativity. Chen [8] (see [9]) introduced the concept CR-warped product submanifolds of Kaehler manifolds. We study CR-slant warped product submanifolds of nearly cosymplectic manifolds which are the more general classes of contact metric manifolds.
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