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We continue studying the σ -Ricci vector field u on a Riemannian manifold (N m , g) , which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold (N m , g) , m > 1 , of positive scalar curvature τ , admits a closed σ -Ricci vector field u such that the vector u − ∇ σ is an eigenvector of T with eigenvalue τ m − 1 , if and only if it is isometric to the m-sphere S α m . In the second result, we show that if a compact and connected T-manifold (N m , g) , m > 2 , admits a σ -Ricci vector field u with σ ≠ 0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature R i c u , u that has a suitable lower bound, then (N m , g) is isometric to the m-sphere S α m , and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (N m , g) admits a σ -Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature R i c u , u has a lower bound depending on a positive constant α , if and only if (N m , g) is isometric to the m-sphere S α m . [ABSTRACT FROM AUTHOR]

We aim to examine the influence of the existence of a nonzero eigenvector ζ of the de-Rham operator Γ on a k-dimensional Riemannian manifold (N k , g) . If the vector ζ annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field ζ on (N k , g) , there are two operators T ζ and Ψ ζ associated with ζ , called the basic operator and the associated operator of ζ , respectively. We show that the existence of an eigenvector ζ of Γ on a compact manifold (N k , g) , such that the integral of Ric (ζ , ζ) admits a certain lower bound, forces (N k , g) to be isometric to a k-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field ζ on a connected and complete Riemannian space (N k , g) , having div ζ ≠ 0 and annihilating the associated operator Ψ ζ , forces (N k , g) to be isometric to the k-dimensional Euclidean space, provided that the squared length of the covariant derivative of ζ possesses a certain lower bound. [ABSTRACT FROM AUTHOR]

We use a nontrivial concircular vector field u on the unit sphere S n + 1 in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere S n + 1 naturally inherits a vector field v and a smooth function ρ. We use the condition that the vector field v is an eigenvector of the de-Rham Laplace operator together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find a characterization of small spheres in the unit sphere S n + 1 . We also use the condition that the function ρ is a nontrivial solution of the Fischer–Marsden equation together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find another characterization of small spheres in the unit sphere S n + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we establish two sharp inequalities for the normalized δ-Casorati curvatures of submanifolds in a Kenmotsu space form, tangent to the structure vector field of the ambient space.Moreover, we show that in both cases the equality at all points characterizes the totally geodesic submanifolds. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kähler product manifold. [ABSTRACT FROM AUTHOR]

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound. [ABSTRACT FROM AUTHOR]