Search Results (49)

The main objective of this paper is to investigate the properties related to the sectional curvatures of a Kähler golden manifold, an almost Hermitian golden manifold whose almost complex golden structure is parallel with respect to the Levi–Civita connection. Under certain conditions, we prove that a Kähler golden manifold with constant sectional curvature is flat. We introduce the concepts of $\Phi$-holomorphic sectional curvature and $\Phi$-holomorphic bi-sectional curvature on a Kähler golden manifold, and compare them respectively with the holomorphic sectional curvature and holomorphic bi-sectional curvature on a Kähler manifold. Full article

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak $\beta$-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak $\beta$-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature. Full article

This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. Full article

In this paper, we prove that the complex four dimensional compact holomorphic symplectic manifold we found earlier is not formal. This gives another strong consequence that it is not a topological Kähler manifold. We also conjecture that this is true for the higher dimensional ones. Full article

We present novel implementations of Starobisky-like inflation within supergravity adopting Kähler potentials for the inflaton which parameterizes hyperbolic geometries known from T-model inflation. The associated superpotentials are consistent with an R and a global or gauge $U{\left(1\right)}_X$ symmetries. The inflaton is represented by a gauge-singlet or non-singlet superfield and is accompanied by a gauge-singlet superfield successfully stabilized thanks to its compact contribution into the total Kähler potential. Keeping the Kähler manifold intact, a conveniently violated shift symmetry is introduced which allows for slight variation in the predictions of Starobinsky inflation: The (scalar) spectral index exhibits an upper bound which lies close to its central observational value whereas the constant scalar curvature of the inflaton-sector Kähler manifold increases with the tensor-to-scalar ratio. Full article

The concept of pointwise slant submanifolds of a Kähler manifold was presented by Chen and Garay. This research extends this notion to a more general setting, specifically in a locally conformal Kähler manifold. We study the pointwise pseudo-slant warped products of the form ${\mathsf{\Sigma }}^{\theta }{\times \hspace{0.17em}}_f{\mathsf{\Sigma }}^{\perp }$ in a locally conformal Kähler manifold. Using the concept of pointwise pseudo-slant, we establish the necessary and sufficient condition for it to be characterized as a warped product submanifold. In addition, we derive several results on pointwise pseudo-slant warped products that expand previously proven main ones. Further, some examples of such submanifolds and their warped products are also given. Full article

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties. Full article

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on ${\mathrm{S}}^6$ was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on ${\mathrm{S}}^6$. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem. Full article

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique $\eta$-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. Full article

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds. Full article

We highlight some properties of a class of distinguished vector fields associated to a $(1,1)$-tensor field and to an affine connection on a Riemannian manifold, with a special view towards the Ricci vector fields, and we characterize them with respect to statistical, almost Kähler, and locally product structures. In particular, we provide conditions for these vector fields to be closed, Killing, parallel, or semi-torse forming. In the gradient case, we give a characterization of the Euclidean sphere. Among these vector fields, the Ricci and torse-forming-like vector fields are particular cases. Full article

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with a pair of pseudo-Riemannian metrics that are mutually associated with each other using the tensor structure. Here, we consider a special class of these manifolds, namely those of the Sasaki-like type. They have an interesting geometric interpretation: the complex cone of such a manifold is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The basic metric on the considered manifold is specialized here as a soliton, i.e., has an additional curvature property such that the metric is a self-similar solution to an intrinsic geometric flow. Almost solitons are more general objects than solitons because they use functions rather than constants as coefficients in the defining condition. A $\beta$-Ricci–Bourguignon-like almost soliton ($\beta$ is a real constant) is defined using the pair of metrics. The introduced soliton is a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein) which, in principle, arise from a single metric rather than a pair of metrics. The soliton potential is chosen to be pointwise collinear to the Reeb vector field, or the Lie derivative of any B-metric along the potential to be the same metric multiplied by a function. The resulting manifolds equipped with the introduced almost solitons are characterized geometrically. Suitable examples for both types of almost solitons are constructed, and the properties obtained in the theoretical part are confirmed. Full article

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification $G^{\mathbf{C}}$ of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. Full article

The aim of the paper is to present two results concerning real hypersurfaces in the six-dimensional sphere $S^6\left(1\right)$. More precisely, we prove that real hypersurfaces with the Lie-parallel shape operator A must be totally geodesic hyperspheres. Additionally, we classify real hypersurfaces in a nearly Kähler sphere $S^6\left(1\right)$ whose Lie derivative of the shape operator coincides with its covariant derivative. Full article