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In this article, we study sequential warped product submanifolds of Sasakian manifolds and provide nonexistence results when the sequential warped product contains one invariant, one pointwise slant and one anti-invariant submanifold. For the only two possible types, we deduce some properties and establish Chen’s inequality in these cases. [ABSTRACT FROM AUTHOR]

We study Ricci solitons on Golden Riemannian manifolds(GRM) with constant sectional curvatures. We define and study Golden Ricci soliton on Golden Riemannian manifolds(GRM). Further, we study Ricci soliton and Golden Ricci soliton on GRM with potential vector fields as concurrent, Killing, conformal Killing, and homothetic. Also, we provide examples of Ricci soliton and Golden Ricci soliton on GRM. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the de Rham cohomology of semi-slant submanifolds in metallic Riemannian manifolds. Some necessary conditions for a semi-slant submanifold of metallic Riemannian manifolds are given to define a well-defined canonical de Rham cohomology class. Also, the non-triviality of such a cohomology class is discussed. Finally, an example is constructed to illustrate the main idea of the paper. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce new class of Riemannian maps which are called pointwise slant Riemannian maps (briefly, PSRM ) as a natural generalization of pointwise slant submanifolds which were introduced by Chen and Garay (Turkish J Math 36:630–640, 2012) and pointwise slant submersions which were defined by Lee and Şahin (Bull Korean Math Soc 51(4):1115–1126, 2014]. We mention many examples and investigate the geometry of foliations which are arisen from the definition of a PSRM and obtain decomposition theorems using the existence of PSRM . Moreover, we find necessary and sufficient conditions for PSRM to be totally geodesic and investigate the harmonicity of such maps. Finally, we get two inequalities in terms of the second fundamental form of a PSRM and check the equality case. [ABSTRACT FROM AUTHOR]

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of C P m . We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric. [ABSTRACT FROM AUTHOR]

In this article, we present a new tensor that generalizes the conharmonic curvature tensor, termed the ψ-conharmonic curvature tensor. In the first part, we derive some fundamental geometrical properties of a pseudo ψ-conharmonically symmetric manifold (briefly,) and give some fascinating outcomes. Among others, we establish that an Einstein manifold is of zero scalar curvature. Besides, we investigate ψ-conharmonically flat perfect fluid and spacetimes, respectively. As a result, we acquire some significant theorems. Finally, we construct several non-trivial examples to establish the existence of. [ABSTRACT FROM AUTHOR]

The present paper attempts to study the almost Ricci solitons in a generalized D-conformally deformed (LCS) n -manifold considering potential vector field as a solenoidal vector field, a gradient vector field or the Reeb vector field of the deformed structure and explicitly investigate the Ricci and scalar curvatures for some cases. We also determine some inequalities for the Ricci curvature of the deformed (LCS) n -manifold when it admits a gradient almost Ricci soliton. [ABSTRACT FROM AUTHOR]

Let G k , n be the Grassmannian of oriented subspaces of dimension k of R n with its canonical Riemannian metric. We study the energy of maps assigning to each P ∈ G k , n a unit vector normal to P. They are sections of a sphere bundle E k , n 1 over G k , n . The octonionic double and triple cross products induce in a natural way such sections for k = 2 , n = 7 and k = 3 , n = 8 , respectively. We prove that they are harmonic maps into E k , n 1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J P on P ⊥ to each P ∈ G 2 , 8 . We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G 2 , 8 . This generalizes the harmonicity of the canonical almost complex structure of S 6 . [ABSTRACT FROM AUTHOR]

The present article deals with the study of generalized curvature equality and inequality. We first derive equality involving mean curvature vector and second fundamental form for sequential warped product submanifolds of a nearly Kähler manifold and then present Lawson-Simons type inequality (Lawson and Simons, Ann. of Math. 98:427–450, 1973) in generalized complex space form that extends the result derived by Sahin in Sahin (Period Math. Hung., 2021). Meanwhile, some special cases and a numerical example in support of inequality are also discussed. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere S n (r) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with α ≠ 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space E n or Euclidean sphere S n. Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results. [ABSTRACT FROM AUTHOR]

The prime object in this article is to study an almost η-Ricci–Yamabe soliton and gradient almost η-Ricci–Yamabe soliton within the framework of almost Kenmotsu manifolds. It is shown that normal almost Kenmotsu manifold admitting an almost η-Ricci–Yamabe soliton or gradient η-Ricci–Yamabe soliton is locally isometric to hyperbolic space ℍ2n+1(−1). Next, we prove that if a (κ, μ) almost Kenmotsu manifold admits an almost η-Ricci–Yamabe soliton, then the manifold is η-Einstein. Besides, we find the condition for non-normal almost Kenmotsu manifolds acknowledging gradient almost η-Ricci–Yamabe soliton. Moreover, an almost η-Ricci–Yamabe soliton on (κ, μ)′-almost Kenmotsu manifold has been studied. Lastly, we construct an example of a gradient almost η-Ricci–Yamabe soliton on a 3-dimensional Kenmotsu manifold. [ABSTRACT FROM AUTHOR]

The objective of this paper is to apply the notion of D - ho-mothetic deformation to almost paracontact almost paracomplex Riemannian manifolds. In case the manifold is para-Sasaki-like, the relation between their Levi-Civita connections is given. It is shown that the D - homothetic deformation of para-Einstein-like manifold and para-Ricci-like soliton is also a para-Einstein-like manifold and para-Ricci-like soliton, respectively. Moreover, if para-Sasaki-like manifold M admits a gradient almost Ricci-like soliton, it is investigated whether the D - homothetic deformation of M admits a gradient Ricci-like soliton. Finally, it is given an example. [ABSTRACT FROM AUTHOR]

In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article. [ABSTRACT FROM AUTHOR]

In this paper, we construct a geometrical compactification of the geodesic flow of non-compact complete hyperbolic surfaces Σ without cusps having finitely generated fundamental group. We study the dynamical properties of the compactified flow, for which we show the existence of attractive circles at infinity. The geometric structure of T 1 Σ for which this compactification is realized is the pair of one-dimensional distributions tangent to the stable and unstable horocyles of T 1 Σ . This is a Kleinian path structure, that is a quotient of an open subset of the flag space by a discrete subgroup Γ of PGL 3 (R) . Our study relies on a detailed description of the dynamics of PGL 3 (R) on the flag space, and on the construction of an explicit fundamental domain for the action of Γ on its maximal open subset of discontinuity in the flag space. [ABSTRACT FROM AUTHOR]

We compute almost-complex invariants h∂¯p,oh_{\bar \partial }^{p,o}, hDolp,oh_{Dol}^{p,o} and almost-Hermitian invariants hδ¯p,oh_{\bar \delta }^{p,o} on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a compatible symplectic structure on a given compact almost-complex manifold. Notice that, when (X, J, g, ω) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h∂¯p,qh_{\bar \partial }^{p,q}.

We prove that if an η\eta -Einstein para-Kenmotsu manifold admits a conformal η\eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η\eta -Ricci soliton is Einstein if its potential vector field VV is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η\eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η\eta -Ricci soliton and satisfy our results. We also have studied conformal η\eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

We investigate the Fu-Yau equation on compact almost astheno-Kähler manifolds and show an a priori C0-estiamte for a smooth solution of the equation.

In this paper, we study conformal hemi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds as a generalized case of conformal semi-invariant submersions and conformal slant submersions. In this manner, we obtain some results of integrability of the distributions which are arisen from the definition of the submersions and study the geometry of the leaves of distributions. Moreover, we obtain the conditions for the totally geodesicness of the submersion. Also, we give an example of proper conformal hemi-slant submersion from almost product manifold onto Riemannian manifold. [ABSTRACT FROM AUTHOR]

In this paper, we introduce hsR maps and ahsR maps from almost quaternionic Hermitian manifolds into Riemannian manifolds as a generalization of almost h-conformal semi-invariant submersions and conformal semi-invariant Riemannian maps. We give some examples of these maps and study their properties: the geometry of foliations, the integrability of distributions, the conditions for these maps to be harmonic or totally geodesic. [ABSTRACT FROM AUTHOR]

We show the short-time existence of the second Ricci flow on a complete noncompact almost Hermitian manifold, and prove that along the second Ricci flow, the non-positivity of the first Chern–Ricci curvature can be preserved if the initial almost Hermitian metric has non-positive bisectional curvature. If additionally the first Chern–Ricci curvature of the initial metric is negative at least at one point, then we show that the almost complex structure of a complete noncompact non-quasi-Kähler almost Hermitian manifold equipped with such a metric cannot be integrable. [ABSTRACT FROM AUTHOR]

Typical examples of uniform magnetic fields are Kähler magnetic fields on Kähler manifolds. It is very difficult to study magnetic Jacobi fields of non-uniform magnetic fields in an arbitrary Riemannian manifold. The canonical magnetic fields of Sasakian manifolds are non-uniform but exact. In this paper, we completely determine magnetic Jacobi fields on Sasakian space forms of dimension greater or equal to 5. [ABSTRACT FROM AUTHOR]

We consider, for n ⩾ 3 , K-quasiregular vol N × -curves M → N of small distortion K ⩾ 1 from oriented Riemannian n-manifolds into Riemannian product manifolds N = N 1 × ⋯ × N k , where each N i is an oriented Riemannian n-manifold, and the calibration vol N × ∈ Ω n (N) is the sum of the Riemannian volume forms vol N i of the factors N i of N. We show that, in this setting, K-quasiregular curves of small distortion are carried by quasiregular maps. More precisely, there exists K 0 = K 0 (n , k) > 1 having the property that, for 1 ⩽ K ⩽ K 0 and a K-quasiregular vol N × -curve F = (f 1 , ... , f k) : M → N 1 × ⋯ × N k , there exists an index i 0 ∈ { 1 , ... , k } for which the coordinate map f i 0 : M → N i 0 is a quasiregular map. As a corollary, we obtain first examples of decomposable calibrations for which corresponding quasiregular curves of small distortion are discrete and admit a version of Liouville's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study Golden-like statistical manifolds. We obtain some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provide a couple of examples.

We characterize the k-th Gauduchon condition and by applying its characterization, we reprove that a compact k-th Gauduchon, semi-Kähler manifold becomes quasi-Kähler, which tells us that in particular, a compact almost pluriclosed, semi-Kähler manifold is quasi-Kähler.

In this paper, we establish some relationships between the main intrinsic invariants, scalar and Ricci curvatures, and the main extrinsic invariant, the mean curvature vector, for slant submanifolds of S-space-forms. In addition to that, we study those slant submanifolds satisfying the equality case between the above invariants, due to the great importance that these submanifolds have had in both complex and contact geometry. Moreover, we present some examples on these conditions. [ABSTRACT FROM AUTHOR]

Real hypersurfaces in a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space) are interesting objects among submanifolds in Riemannian manifolds. It is known that a real hypersurface in a nonflat complex space form admits an almost contact metric structure (ϕ , ξ , η , g) induced from the ambient space. Hence we are interested in real hypersurfaces from the aspects of both submanifolds and almost contact metric manifolds. In this paper, we study real hypersurfaces in a nonflat complex space form from the viewpoint of a recurrence of the tensor field h (= (1 / 2) L ξ ϕ) . We note that the tensor h plays an important role in contact Riemannian geometry. We give a new classification which includes a special class of 3-dimensional ruled real hypersurfaces in a complex hyperbolic plane C H 2 (c) . [ABSTRACT FROM AUTHOR]

In this paper, we study the biwarped product submanifold in locally nearly metallic Riemannian manifold. We construct a non trivial example of a biwarped product submanifold in metallic Riemannian manifold. Moreover, we discuss a necessary and sufficient condition for such submanifolds to be locally trivial. Finally, we set up an inequality in locally nearly metallic Riemannian manifold for the second fundamental form with respect to some conditions. We also investigate the equality case. [ABSTRACT FROM AUTHOR]

In this article, we study the problem of the existence and nonexistence of warping function associated with constant scalar curvature on pseudo-Riemannian Poisson warped product space under the assumption that fiber space has constant scalar curvature. We characterize the warping function on Einstein Poisson warped space by taking the various dimensions of base space B (i.e; (1). d i m B = 1 , (2). d i m B ≥ 2 ). [ABSTRACT FROM AUTHOR]

In this paper, we introduce left-invariant (strict) nearly para-Kaähler structures on Lie groups (nearly para-Kähler Lie algebras) and present some properties of them. We prove the existence of a unique connection in terms of the characteristic connection on these Lie groups with totally skew-symmetric torsion tensor and we show that the Nijenhuis tensor is parallel with respect to the characteristic connection. We determine conditions that allow the curvature tensor of the characteristic connection satisfies the first Bianchi identity. Finally, we focus on anti-abelian almost para-complex structures on Lie groups and we study nearly para-Kähler conditions for these structures. In this study we encounter bi-invariant metrics as pseudo-Riemannian metrics that accept nearly para-Kähler structures. Moreover, we present some examples of nearly para-Kähler Lie algebras. [ABSTRACT FROM AUTHOR]

Recently, trans-S-manifolds have been defined as a wide class of metric f-manifolds which includes, for instance, f-Kenmotsu manifolds, S-manifolds and C-manifolds and generalize well-studied trans-Sasakian manifolds. The definition of trans-S-manifolds is formulated using the covariant derivative of the tensor f and although this formulation coincides with the characterization of trans-Sasakian manifolds in such a particular case, this latter type of manifolds were not initially defined in this way but using the Gray-Hervella classification of almost Hermitian manifolds. The aim of this paper is to study how (almost) trans-S-manifolds relate with the Gray-Hervella classification and to establish both similarities and differences with the trans-Sasakian case. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce screen generic lightlike submanifold of golden semi-Riemannian manifolds and give non-trivial examples of such submanifolds. We obtain necessary and sufficient conditions for integrability of the distributions. Further, we find a necessary and sufficient condition for the induced connection to be a metric connection. We also obtain necessary and sufficient condition for totally umbilical screen generic lightlike submanifold of golden semi-Riemannian manifolds to be totally geodesic. Finally, we find some necessary and sufficient conditions for minimal screen generic lightlike submanifold of golden semi-Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2 n ≥ 10 (respectively ≥ 6 ) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds). [ABSTRACT FROM AUTHOR]

Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, the notions of almost Hermitian, almost Kähler statistical manifolds, and also some meaningful classes of almost contact metric statistical manifolds are introduced. Then the almost contact metric, contact metric and almost cosymplectic statistical manifolds are studied as real hypersurfaces in almost Hermitian and almost Kähler statistical manifolds. Examples of almost Hermitian and almost Kähler statistical manifolds and their statistical hypersurfaces are given. [ABSTRACT FROM AUTHOR]

We introduce the conformally balanced condition on almost Hermitian manifolds. We show that a compact Kähler-like and G-Kähler-like almost Hermitian manifold with the conformally balanced condition becomes Kähler. [ABSTRACT FROM AUTHOR]

Let (ω , λ) be a stable Hamiltonian structure on a closed oriented manifold M of dimension 2 n - 1 , F the stable Hamiltonian foliation, generated by the Reeb vector field R of (ω , λ) , and H B k (M , F) , the kth basic cohomology group of (M , F) ; see Section 1 for definitions. In this paper, we give some topological properties of (ω , λ) . In particular, we prove the following results: For all k ∈ { 1 , ... , n - 1 } , 0 ≠ [ ω k ] ∈ H B 2 k (M , F) , which allows us to give an example of a manifold with a Hamiltonian structure, which is not stable. If dim H B 2 (M , F) = 1 , then M is a co-symplectic manifold (symplectic mapping torus), or contact manifold. [ABSTRACT FROM AUTHOR]