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Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of almost contact metric manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and weak nearly Sasakian manifolds admit two types of totally geodesic foliations. Our main result generalizes the theorem by Cappelletti-Montano and Dileo (Ann Matem Pura Appl 195:897-922, 2016) to the context of weak almost contact geometry. [ABSTRACT FROM AUTHOR]

In this paper we study isometric immersions f : M n → C ′ P n of an n-dimensional pseudo-Riemannian manifold M n into the n-dimensional para-complex projective space C ′ P n . We study the immersion f by means of a lift f of f into a quadric hypersurface in S n + 1 2 n + 1 . We find the frame equations and compatibility conditions. We specialize these results to dimension n = 2 and a definite metric on M 2 in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the classifications of β -Kenmotsu statistical structures of constant ϕ -sectional curvature on β -Kenmotsu space forms. Our results show that such β -Kenmotsu statistical structures on a β -Kenmotsu space form with dimension greater than 3 must be almost-trivial; on a 3-dimensional β -Kenmotsu space form, in addition to the almost-trivial β -Kenmotsu statistical structure, there exist other β -Kenmotsu statistical structures which satisfy the constant ϕ -sectional curvature condition. These results generalize and improve the corresponding results for Kenmotsu statistical manifolds Jiang et al. (Hacet J Math Stat 51(3):800–816, 2022) as well as cosymplectic statistical manifolds Yan et al. (J Geom Phys 196:105076, 2024). Before obtaining these main results on β -Kenmotsu statistical structures, we also obtain the expression for the curvature tensor of the β -Kenmotsu space form in terms of Riemannian geometry, and many examples are given by using warped products. [ABSTRACT FROM AUTHOR]

For an arbitrary real nilpotent Lie algebra (nilmanifold) with an integrable complex structure, we propose an algorithm for constructing a special model of this nilmanifold that includes information on the complex structure. As a main application, we obtain a classification of eight-dimensional -generated nilpotent Lie algebras that admit an integrable complex structure. We also describe the moduli spaces of complex structures for each Lie algebra from the resulting classification list. [ABSTRACT FROM AUTHOR]

In this paper, we have examined the idea of pointwise quasi bi-slant submanifolds of Kenmotsu manifolds and studied warped product pointwise quasi bi-slant submanifolds of Kenmotsu manifolds. We obtain several results on pointwise quasi bi-slant submanifolds of Kenmotsu manifolds and proved the characterization theorem for warped product pointwise quasi bi-slant submanifolds. Also, we provide a non-trivial example of such warped product submanifold. [ABSTRACT FROM AUTHOR]

On a real hypersurface MM in complex projective space, we can define two tensor fields of type (1, 2), AF(k){A}_{F}^{\left(k)} and AT(k){A}_{T}^{\left(k)}, associated with the shape operator AA of the real hypersurface, for any nonnull real number kk. Following Tachibana [Analytic tensor and its generalization, Tohoku Math. J. 12 (1960), 208–221], we study the purity and hybridness of such tensors with respect to AA.

In this paper, we first consider the 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold. [ABSTRACT FROM AUTHOR]

Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra, the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generalization of the complete lifts. [ABSTRACT FROM AUTHOR]

We prove that every compact, homogeneous η -Einstein almost cosymplectic manifold is a cosymplectic manifold. [ABSTRACT FROM AUTHOR]

This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete K-contact or Sasakian manifold endowed with a closed GRA soliton satisfying 4 c 1 c 2 ≠ 1 is compact Einstein with scalar curvature 2 n (2 n + 1) . As for the gradient case, it exhibits an isometry to the unit sphere S 2 n + 1 . Subsequently, we identify a few adequate conditions under which a non-trivial complete K-contact manifold with a GRA soliton is trivial (η -Einstein). Following that, we establish certain results on H-contact and complete contact manifolds. We also demonstrate that a non-Sasakian (k , μ) -contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle R n + 1 × S n (4) , provided 4 c 1 c 2 (1 - 2 n) ≠ 1 and c 2 ≠ 0 . Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime. [ABSTRACT FROM AUTHOR]

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that there is no compact complex surface which is homotopy equivalent to a closed real-hyperbolic 4-manifold. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures. [ABSTRACT FROM AUTHOR]

In the present paper, we initiate the study of ∗ - η -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a ∗ - η -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost ∗ - η -Ricci soliton. Next, we deliberate ∗ - η -Ricci soliton admitting (κ , μ) ′ -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to H n + 1 (- 4) × R n . Finally we present some examples to decorate the existence of ∗ - η -Ricci soliton, gradient almost ∗ - η -Ricci soliton on Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds. [ABSTRACT FROM AUTHOR]

We show that every quasiconformal contact foliation supports an invariant metric and characterise these foliations by the dynamical property of C 1 -equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for quasiconformal contact foliations, and provide a lower bound to the number of closed leaves. In particular, we show that the Weinstein conjecture holds for quasiconformal Reeb fields. [ABSTRACT FROM AUTHOR]

In the present article, we extend the notion of vacuum static equations on almost coKähler manifolds and rename them as nearly vacuum static equations. It is shown that if an η -Einstein almost coKähler manifold admits a non-trivial solution of a nearly vacuum static equation, then the solution must be a constant. In (κ , μ) -almost coKähler manifolds, the non-trivial solutions of nearly vacuum static equations do not exist. We also apply nearly vacuum static equations on perfect fluid spacetimes as well as generalized Robertson–Walker spacetimes. Among others, it is shown that a perfect fluid spacetime admitting nearly vacuum static equations is of constant scalar curvature and a generalized Robertson–Walker spacetime obeying nearly vacuum static equations represents a dark matter era. [ABSTRACT FROM AUTHOR]

The present paper is designed to study the geometric composition of an n-dimensional Sasakian manifold with ∗ -Ricci–Bourguignon soliton under Zamkovoy connection. Here, we have shown the characteristics of the soliton when it is satisfied by the metric of the Sasakian manifold with respect to Riemannian connection and also with respect to Zamkovoy connection. We have also acquired Laplace equation from the soliton equation when the potential vector field V of the soliton is of gradient type in terms of Zamkovoy connection. We have then studied some well known curvature tensor such as W 2 curvature tensor and Q curvature tensor on the manifold when it admits the ∗ -Ricci–Bourguignon soliton with respect to Zamkovoy connection. [ABSTRACT FROM AUTHOR]

It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures (J a , b , g a , b) . We show in this article that the complex structure J a , b is harmonic with respect to g a , b , i.e., it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon ( k ≥ 2 ). Finally, we study the Bismut connection associated to (J a , b , g a , b) and we provide formulas for the Bismut-Ricci tensor Ric B and the Bismut-Ricci form ρ B . We show that these tensors vanish if and only if each Sasakian factor is η -Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds. [ABSTRACT FROM AUTHOR]

In this paper we study some curves of a trans-Sasakian manifold that generalize magnetic curves. [ABSTRACT FROM AUTHOR]

An almost abelian Lie group is a solvable Lie group with a codimension one normal abelian subgroup. We characterize almost Hermitian structures on almost abelian Lie groups where the almost complex structure is harmonic with respect to the Hermitian metric. Also, we adapt the Gray–Hervella classification of almost Hermitian structures to the family of almost abelian Lie groups. We provide several examples of harmonic almost complex structures in different Gray–Hervella classes on some associated compact almost abelian solvmanifolds. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

We investigate the exterior differential operator on quasi-Kähler manifolds and show some relations of its components for smooth functions. [ABSTRACT FROM AUTHOR]

The idea of potentially infinite metrics on space-time was given by Bennett Chow and Sun-Chin Chu in 1995. After that Perelman introduced two new ideas about such metric in 2002. In this article, we use this idea in space-time warped product manifold and study gradient Ricci soliton space-time warped product. First, we introduce the notion of space-time warped product, P = (B × I) × f F with potentially infinite metric, G = (g B + (R + N 2 t) d t 2) + f 2 g F. Then, we investigate the behavior of Ricci curvature tensor up to O (N - 1) and discuss potentially gradient Ricci soliton for space-time warped product. Next, we prove existence conditions for gradient Ricci soliton space-time warped product. Then, we discuss several results for shrinking, steady, or expanding gradient Ricci soliton (P , G , ∇ ϕ , λ) , with compact base and fiber with at least dimension two. After that, we emphasize the construction of class of expanding Ricci soliton. Furthermore, we discuss the compactness of space-time manifold when space-time warped product satisfies some inequality. We also give examples of generalized black holes, which metric can be written as space-time warped product with potentially infinite metric. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study statistical submersions with parallel almost complex structures. First, we define the notion of the generalized Kähler-like statistical submersion and give examples of the Kähler-like statistical submersions. In addition, we investigate total space and fibers under certain conditions. After, we introduce some results on J-invariant, J ∗ -invariant and anti-invariant generalized Kähler-like statistical submersions. [ABSTRACT FROM AUTHOR]

In this paper, we study slant Riemannian submersions from conformal Sasakian manifolds onto Riemannian manifolds. We investigate the geometry of foliations associated with the submersion and give sufficient conditions for the submersion to be harmonic. [ABSTRACT FROM AUTHOR]

We study Chern flat, Chern–Ricci flat twisted product almost Hermitian manifolds. We extend the twisted product to almost Hermitian manifolds, and give some formulae of the Chern curvature, the Chern–Ricci curvature and the Chern scalar curvature on the twisted product almost Hermitian manifold under the quasi-Kähler condition. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to analyze sharp type inequalities including the scalar and Ricci curvatures of anti-invariant Riemannian submersions in Kenmotsu space forms K s (ε) . We give non-trivial examples for anti-invariant Riemannian submersions, investigate some curvature relations between the total space and fibres according to vertical and horizontal cases of ξ . Moreover, we acquire Chen-Ricci inequalities on the ker ϑ ∗ and (ker ϑ ∗) ⊥ distributions for anti-invariant Riemannian submersions from Kenmotsu space forms according to vertical and horizontal cases of ξ . [ABSTRACT FROM AUTHOR]

In this paper, we prove the non-existence of non-trivial statistical structures of constant holomorphic sectional curvature based on complex space forms with dimension greater than 2. For 2-dimensional complex space forms we show an example to illustrate there do exist non-trivial statistical structures of constant holomorphic sectional curvature, and we also obtain a rigidity theorem in this case. Finally, in contrast to complex space forms, we construct some new examples of non-trivial statistical structures of constant sectional curvature based on real space forms. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of a pseudo-Kähler structure on the deformation space B 0 (T 2) of properly convex R P 2 -structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of B 0 (T 2) with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichmüller space. We show that the S 1 -action on B 0 (T 2) , given by rotation of the fibers, is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the SL (2 , R) -action over B 0 (T 2) . [ABSTRACT FROM AUTHOR]

We study geodesics and magnetic trajectories in the model space F4{{\rm{F}}}^{4}. The space F4{{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F4{{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F4{{\rm{F}}}^{4}. Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F4{{\rm{F}}}^{4}. We explore some typical submanifolds of F4{{\rm{F}}}^{4}. Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F4{{\rm{F}}}^{4}.

Abstract The aim of the present paper is to analyze sharp type inequalities including the scalar and Ricci curvatures of anti-invariant Riemannian submersions in Kenmotsu space forms $$K_{s}(\varepsilon )$$ K s ( ε ) . We give non-trivial examples for anti-invariant Riemannian submersions, investigate some curvature relations between the total space and fibres according to vertical and horizontal cases of $$\xi $$ ξ . Moreover, we acquire Chen-Ricci inequalities on the $$\ker \vartheta _{*}$$ ker ϑ ∗ and $$(\ker \vartheta _{*})^{\bot }$$ ( ker ϑ ∗ ) ⊥ distributions for anti-invariant Riemannian submersions from Kenmotsu space forms according to vertical and horizontal cases of $$\xi $$ ξ .

We provide some properties of Riemann solitons with torse-forming potential vector fields, pointing out their relation to Ricci solitons. We also study those Riemann soliton submanifolds isometrically immersed into a Riemannian manifold endowed with a torse-forming vector field, having as potential vector field its tangential component. We consider the minimal and the totally geodesic cases, too, as well as when the ambient manifold is of constant sectional curvature. In particular, we prove that a totally geodesic submanifold isometrically immersed into a Riemannian manifold endowed with a concircular vector field is a Riemann soliton if and only if it is of constant curvature. Furthermore, we show that, if the potential vector field of a minimal hypersurface Riemann soliton isometrically immersed into a Riemannian manifold of constant curvature and endowed with a concircular vector field is of constant length, then it is a metallic shaped hypersurface. [ABSTRACT FROM AUTHOR]

In this paper we give in the context of sections of a (Riemannian) almost Hermitian fiber bundle a reformulation of a theory of deformation linked to harmonic maps between almost Hermitian manifolds developped by A. Lichnerowicz. [ABSTRACT FROM AUTHOR]

Pasquier and Perrin discovered that the G2-horospherical manifold X of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric; in other words, it can be deformed nontrivially to the rational homogeneous space. We show that X is the only smooth projective variety with this property. This is obtained as a consequence of our main result that X can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to X if and only if its VMRT at a general point is projectively isomorphic to that of X. We employ the method the authors developed to solve the corresponding problem for symplectic Grassmannians, which constructs a flat Cartan connection in a neighborhood of a general minimal rational curve. In adapting this method to X, we need an intricate study of the positivity/negativity of vector bundles with respect to a family of rational curves, which is subtler than the case of symplectic Grassmannians because of the nature of the differential geometric structure on X arising from VMRT. [ABSTRACT FROM AUTHOR]

The present research article deals with the study of almost η-Ricci-Bourguignon soliton and gradient almost η-Ricci-Bourguignon soliton on almost Kenmotsu manifolds. It is shown that if the metric of a Kenmotsu manifold M2n+1 admits a gradient almost η-Ricci-Bourguignon soliton, then it is η-Einstein. More-over, if the manifold is complete and ξ leaves the scalar curvature invariant, then it is locally isometric to Hyperbolic space ℍ2n+ 1(−1). Next, we demonstrate that if a (κ, µ) almost Kenmotsu manifold admits an almost η-Ricci-Bourguignon soliton, then the manifold is η-Einstein. Besides, we explore the condition for non-normal almost Kenmotsu manifolds satisfying gradient almost η-Ricci-Bourguignon soliton. In addition, we have also investigated an almost η-Ricci-Bourguignon soliton on (κ, µ)′-almost Kenmotsu manifold. [ABSTRACT FROM AUTHOR]