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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]

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ -invariant, δ (2) , and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen's first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades. [ABSTRACT FROM AUTHOR]

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities and provide illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized δ -Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ -sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example. [ABSTRACT FROM AUTHOR]

This correction notice addresses an omission in a previously published paper titled "Ricci Vector Fields Revisited." The authors apologize for not including acknowledgements in the original paper and express their gratitude for the support received from King Saud University in Riyadh, Saudi Arabia. The authors assure readers that this correction does not impact the scientific conclusions of the paper. The correction has been approved by the Academic Editor and the original publication has been updated. The authors of the paper are Hanan Alohali, Sharief Deshmukh, and Gabriel-Eduard Vîlcu. [Extracted from the article]

In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kähler-like statistical manifold and derive the main properties of quaternionic Kähler-like statistical submersions, extending in a new setting some previous results obtained by K. Takano concerning statistical manifolds endowed with almost complex and almost contact structures. Finally, we give a nontrivial example and propose some open problems in the field for further research. [ABSTRACT FROM AUTHOR]

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ -invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on δ -invariants and their applications have been summarized in the first author's book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the δ -invariants done mostly after the publication of the book. [ABSTRACT FROM AUTHOR]

In this paper, we find a characterization of the 3-sphere using 3-dimensional compact and simply connected trans-Sasakian manifolds of type (α ,   β) . [ABSTRACT FROM AUTHOR]

The purpose of this article is to establish some inequalities concerning the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided. [ABSTRACT FROM AUTHOR]

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere S m (c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation. [ABSTRACT FROM AUTHOR]

The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it is shown that the geodesicity property of vector fields is also useful in characterizing not only spheres, but also Euclidean spaces. [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]

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the class of quasi-homogeneous production models, obtaining the classification of such models with constant elasticity with respect to an input as well as with respect to all inputs. Moreover, we prove that a quasi-homogeneous production function f satisfies the proportional marginal rate of substitution property if and only f reduces to some symmetric production functions. [ABSTRACT FROM AUTHOR]

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ -sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h ∗ of the submanifold (associated with the dual connections) satisfy h = − h ∗ , i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection. [ABSTRACT FROM AUTHOR]