In this work, we revisit quasi-Sasakian structures in dimension three and examine how these structures interact with the foliation generated by the Reeb vector field and its basic cohomology. Through a deformation-based approach, we show that a closed, orientable 3-manifold admits a quasi-Sasakian structure precisely when it is either Sasakian or arises as a Kähler mapping torus. In particular, every quasi-Sasakian structure in this setting can be deformed into a Sasakian or a co-Kähler one. This result leads to a complete classification of quasi-Sasakian manifolds in dimension three and highlights the geometric and topological features that distinguish the two cases.

We introduce and study locally conformal almost generalized f-cosymplectic manifolds, a new class of almost contact metric structures that generalizes both locally conformal almost cosymplectic and almost f-cosymplectic geometries. Such a structure is determined by a closed Lee form ω and a smooth function f satisfying dη = ω∧η , dΦ = 2fη∧Φ + 2ω∧Φ , where Φ (· ,· ) = g(· ,ϕ· ) is the fundamental 2-form. Our main result reveals a sharp dimensional dichotomy: in dimension 3, ω may be transverse to the contact form η , whereas in dimensions 5 and higher, ω is necessarily proportional to η . This rigidity, which has no analogue in even-dimensional conformal symplectic geometry, is derived from integrability conditions and illustrated by explicit examples in dimensions 3 and 5. The framework provides a unified geometric setting for investigating Reeb foliations, curvature identities, and global properties of almost contact metric manifolds with locally conformal symplectic leaves.

This paper investigates timelike conformal vector fields on closed Lorentzian 3-manifolds and shows that, although these fields form a broader class than Killing fields, their behavior in dimension three is nonetheless remarkably rigid. After performing a conformal change of the metric so that the vector field becomes unit and Killing, we analyze the geometry of the flow it generates through the framework of stable Hamiltonian structures and basic cohomology. Our main result proves that any nowhere-vanishing timelike conformal vector field necessarily arises as the Reeb vector field of either a Sasakian structure or a co-Kähler structure. In other words, every such Lorentzian conformal flow is intrinsically "Reeb-like", which forces the underlying geometry to be either contact or cosymplectic. This establishes a striking connection between Lorentzian geometry, Sasakian and co-Kähler structures, and the topology of flows in dimension 3.

This paper investigates the geometric implications of locally conformal almost cosymplectic structures on (k, μ )' -spaces. We prove that there exist integrable distributions 𝒟_3 and 𝒟_3^⊥ such that locally conformal almost cosymplectic (k, μ )' -manifolds decompose locally as the Riemannian product of a totally geodesic manifold and a 2-dimensional totally geodesic surface with Gaussian curvature -k .

We introduce and study almost cosymplectic manifolds whose characteristic vector field xi belongs to a generalized (k, mu)(y)-nullity distribution, which involves the tensor h(y) = phi o h. If h(y) not equal 0, we prove that the smooth functions k and mu are uniquely determined by k < 0 and mu = xi(ln lambda), where lambda=root-k. Moreover, we show that the spectrum of h(y) is {0, lambda,-lambda} and that the considered manifolds cannot be Ricci symmetric. Under certain conditions, we further prove that such manifolds are locally a warped product. Examples are also provided.

We introduce some geometric properties of a horizontally conformal quasi-hemi-slant Riemannian submersion from a Sasakian manifold, normal to the characteristic vector field, supported by an example. Under some conditions, we obtain geometric configurations of fibres and the base manifold of such submersions. We also give a characterization theorem for the proper horizontal conformal quasi-hemi-slant Riemannian submersions with totally umbilical fibres.

We discuss the geometric impact of a Killing vector field on certain almost contact manifolds with constant sectional curvatures. We prove that the additional structures given in A. De Nicola, G. Dileo, I. Yudin (2018) can be naturally constructed in the case of nearly Sasakian and nearly cosymplectic manifolds with constant sectional curvatures.

In the present article, we consider bi-slant submanifolds in trans-Sasakian generalized Sasakian space forms. Specifically, we establish both the Chen first inequality and the Chen-Ricci inequality on such submanifolds. We provide an example of bi-slant submanifold.

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection is obtained.

We introduce invariant rigged null hypersurfaces of indefinite almost contact manifolds, by paying attention to those of indefinite nearly alpha-Sasakian manifolds. We prove that, under some conditions, there exist leaves of the integrable screen distribution of the ambient manifolds admitting nearly alpha-Sasakian structures.

We consider a class of almost cosymplectic manifolds with Kählerian leaves and η -parallel tensor h'=ϕ∘ h . We prove that there exists a distribution of eigenvalues of h' with totally umbilical leaves and foliations whose leaves are minimal almost Kählerian manifolds immersed as hypersurfaces of almost cosymplectic manifolds. We show that if the spectrum of h' is {0, λ , -λ} , with 0 as simple eigenvalue and λ a positive real number and ∇ _ξh'=0 , the manifolds under consideration are (-λ ^2) -space. Supporting examples are also established.

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant ϕ-holomorphic sectional curvature H=1 or a 5-dimensional proper nearly Sasakian manifold with a constant ϕ-holomorphic sectional curvature H>1. We also prove that the spectrum of the operator H2 generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature.

We introduce a geometric flow on a screen integrable null hypersurface in terms of its local second fundamental form. We use it to give an alternative proof to the vorticity free Raychaudhuri’s equation for null hypersurface, as well as establishing conditions for the existence of constant mean curvature (CMC) null hypersurfaces, and leaves of constant scalar curvatures.

Under a pulled-back approach given in [1] and firstly presented in [2], we introduce, in this paper, the concepts of almost contact and normal almost contact Finsler structures on the pulled-back bundle. Properties of structures partly Sasakians are studied. Using the hh-curvature tensor of Chern connection given in [2], we obtain some characterizations of horizontally Finslerian K-contact structures via the horizontal Ricci tensor and the flag curvature.

Locally, a screen integrable globally null manifold $M$ splits through a Riemannian leaf $M'$ of its screen distribution and a null curve $\mathcal{C}$ tangent to its radical distribution. The leaf $M'$ carries a lot of geometric information about $M$ and, in fact, forms a basis for the study of expanding and non-expanding horizons in black hole theory. In the present paper, we introduce a degenerate Ricci-type flow in $M'$ via the intrinsic Ricci tensor of $M$. Several new gradient estimates regarding the flow are proved.

We investigate a class of locally conformal almost Kähler structures and prove that, under some conditions, this class is a subclass of almost Kähler structures. We show that a locally conformal almost Kähler manifold admits a canonical foliation whose leaves are hypersurfaces with the mean curvature vector field proportional to the Lee vector field. The geodesibility of the leaves is also characterized, and their minimality coincides with the incompressibility of the Lee vector field along the leaves.

Some basic geometric properties of doubly twisted product immersions are established.

Tensors and Foliations of Locally Conformal Almost Kähler Manifolds" [1], there was an error in equation ( 12) in which B � -(1/x 2 2 )X 2 should read as B � -X 2 .

We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [4].

We investigate totally umbilical r-null submanifolds of generalized Robertson-Walker space forms. Using generalized Newton transformations, we obtain new geometric configurations for the mean curvature functions which generalize many well-known results on null geometry.