Abstract
In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in Kenmotsu space forms.
Highlights
In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension
Yano and Ishihara [8] considered a submanifold M whose tangent bundle T M splits into a complex subbundle Ᏸ and a totally real subbundle Ᏸ⊥
Blair and Chen [1] proved that a CR-submanifold of a locally conformal Kaehler manifold is a CauchyRiemann manifold in the sense of Greenfield
Summary
In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. (b) totally real submanifolds, that is, J(TpM) ⊂ Tp⊥M, for all p ∈ M, where TpM (resp., Tp⊥M) is the tangent (resp., the normal) vector space of M at p. Such submanifolds were defined and studied by Chen and Ogiue [4]. A (2m+1)-dimensional Riemannian manifold (M , g) is said to be a Kenmotsu manifold if it admits an endomorphism φ of its tangent bundle T M , a vector field ξ, and a 1-form η, which satisfy: φ2 = − Id +η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0, g(φX, φY ) = g(X, Y ) − η(X)η(Y ), η(X) = g(X, ξ),. P is an endomorphism of tangent bundle T M and F is a normal bundle valued 1-form on T M
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