Let $M$ be a $10$-dimensional closed oriented smooth manifold. Set $$\mathcal{D}_{M} := \{ x \in H^{2}(M; \Z/2) \mid x^{2} + w_{2}(M) x \in ρ_{2} ( TH^{4}(M;\Z) ) \}.$$ Suppose that $H_{1}(M;\Z)=0$ and $\mathcal{D}_{M} \subset ρ_{2}( H^{2}(M; \Z) )$. Then the necessary and sufficient conditions for $M$ to admit a stable almost complex structure are determined in terms of the characteristic classes and cohomology ring of $M$.
We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexisten…
In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le α$, we prove that $\left(\sharp_{i=1}^α M_{i}\right) \sharp (α{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $α$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M\sharp \overline{\mathbb{C} P^{n}}$ also admits an almost complex structure. At last, as an application, we obtain that $α\mathbb{C} P^{2n}\sharp β\overline{\mathbb{C} P^{2n}}$ admits an almost complex structure if and only if $α$ is odd.
In this paper, we introduce generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate this new generalized manifolds with classical almost para-contact manifolds. The paper contains several examples.
In this paper, we show that there is a close relationship between generalized subtangent manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized subtangent maps.
We show the minimal total Betti number of a closed almost complex manifold of dimension $2n\ge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.
In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.
The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(κ,μ)'$-almost Kenmotsu manifolds are constructed to justify our results.
In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kaehler manifolds and a sharp Hessian comparison on nearly Kaehler manifolds.
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of obtained results.
This paper is a study of almost contact statistical manifolds. Especially this study is focused on almost cosymplectic statistical manifolds. We obtained basic properties of such manifolds. It is proved a characterization theorem and a corollary for the almost cosymplectic statistical manifold with Kaehler leaves. We also study curvature properties of an almost cosymplectic statistical manifold. Moreover, examples are constructed.
We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class of $(M, J)$, for the existence of an embedding or an immersion in an almost complex Euclidean $4m$-space. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $R^6$.
We give a condition for an almost constant-type manifold to be a constant-type manifold, and holomorphic and $R$-invariant submanifolds of almost Hermitian manifolds are studied. Generalizations of some results in [5] are given.
In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.
For almost contact metric or almost paracontact metric manifolds there is natural notion of $η$-normality. Manifold is called $η$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $η$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $η$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $η$-normal man…
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds are studied and characterised in terms of the classification used and their geometric properties.
Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an analogue of the idea of slant submanifold in almost Hermitian geometry. There are classified pointwise slant surfaces of four dimensional almost para-Hermitian manifold.
We give the necessary and sufficient conditions for the Penrose-Ward transformation to work on almost $G_2$-manifolds, endowed with natural almost twistorial structures.
As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic, etc. Finally, we give some example…
In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.
Lie groups considered as three-dimensional almost paracontact almost paracomplex Riemannian manifolds are investigated. In each basic class of the classification used for the manifolds under consideration, a correspondence is established between the Lie algebra and the explicit matrix representation of its Lie group.
Relations between conjugate connections with respect to the pair of Norden metrics and to the almost complex structure on almost Norden manifolds are studied. Conjugate connections of the Levi-Civita connections induced by the Norden metrics are obtained. Statistical structures on almost Norden manifolds are considered.
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the…
The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an example by a Lie group.
A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian manifold from a holomorphic complex Riemannian manifold is presented and called an $S^1$-solvable extension.
The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.
This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.
Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional manifolds is constructed and characterized.
In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(κ,μ)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $η$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.
It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. It is proven that the introduced manifold of cosymplectic type is at. Some examples of the studied manifolds are given.
This paper is a complete study of almost α-paracosmplectic manifolds. We characterize almost α-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost α-paracosmplectic manifold are found. We also proved that ξ is a harmonic vector field if and only if it is an eigen vector field of the Ricci operator. We locally classify three dimensional almost α-para-Kenmotsu manifolds satisfying a certain nullity condition. We show that this condition is invariant under D_{γ,β}-homothetic deformation. Furthermore, we construct examples of almost…
In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.
Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered manifolds.
We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.
The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kähler manifold, and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kähler metric vanishes for all …
We show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of $n$-dimensional closed Riemannian manifolds with an almost parallel $p$-form ($2\leq p \leq n/2$) in $L^2$-sense, and give an almost decomposition result of the manifold under some pinching conditions when $2\leq p<n/2$.
Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and suffcient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact metric manifolds are para-CR. We establish necessary and suffcient conditions for paracontact metric manifolds as well as for almost paracosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their conseque…
A four-parametric family of linear connections preserving the almost complex structure is defined on an almost complex manifold with Norden metric. Necessary and sufficient conditions for these connections to be natural are obtained. A two-parametric family of complex connections is studied on a conformal Kähler manifold with Norden metric. The curvature tensors of these connections are proved to coincide.
In the present paper, we introduce the notion of $\ast$-Miao-Tam critical equation on almost contact metric manifolds and studied on a class of almost Kenmotsu manifold. It is shown that if the metric of a $(2n + 1)$-dimensional $(k,μ)'$-almost Kenmotsu manifold $(M,g)$ satisfies the $\ast$-Miao-Tam critical equation, then the manifold $(M,g)$ is $\ast$-Ricci flat and locally isometric to the Riemannian product of a $(n + 1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold. Finally, an illustrative example is presented to support the main theorem.
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spaces, is introduced.
The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.
In this short note, we propose an unified method to derive formulas for derivations conjugated by exponential functions on an almost complex manifold. In v3, we corrected some mistakes in previous versions.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the geometry of the almost complex manifolds with Norden metric. Particularly, we consider the P-connection on a the class of manifolds with nonintegrable almost product structure.
Following T.-J. Li, W. Zhang [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li, W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.] and the results obtained by D. Sullivan in [Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.] t…
We introduce the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion and conformal slant submersions. We investigate several properties of these including the integrability of distributions, the geometry of foliations and the conditions for such maps to be totally geodesic. Further we give some examples of such maps.
An almost Clifford and an almost Cliffordian manifold is a $G$--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group $GL(km, \mathbb R)$ to $GL(m, {\mathcal O})$, where $k=2^{s+t}$ and $m \in \mathbb N$. An almost Cliffordian manifold is given by a reduction of the structure group to $GL(m, \mathcal O) GL(1,\mathcal O)$. We prove that an almost Clifford manifold based on $\mathcal O$ is such that there exists a unique subordinated connection, while the case of an almost Cliffordian…
We investigate new Clairaut conditions for anti-invariant submersions from normal almost contact metric manifolds onto Riemannian manifolds. We prove that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian. Several illustrative examples are also included.
We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.
We show that if a simply connected manifold is almost quarter pinched then it is diffeomorphic to a CROSS (a compact rank one symmetric space) or a sphere.
