This paper is about a generalization of famous Inoue's surfaces. Let $M$ be a matrix in $SL(2n+1,\mathbb{Z})$ having only one real eigenvalue which is simple. We associate to $M$ a complex manifold $T_M$ of complex dimension $n+1$. This manifold fibers over $S^1$ with the fiber $\mathbb{T}^{2n+1}$ and monodromy $M^\top$. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type $T_M$. We prove that if $M$ is not diagonalizable, then $T_M$ does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.
In this paper we construct a family of complex analytic manifolds that generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in $SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic polynomial we associate a manifold $T(M,\mathbf{D})$ (depending on an auxiliary parameter $\mathbf{D}$). This manifold fibers over the $s$-dimensional torus $\mathbb{T}^s$, where $s$ is the number of real eigenvalues of $M$. The fiber is the $N$-dimensional torus $\mathbb{T}^{N}$, and the monodromy matrices are certain polynomials of the matrix $M$. The basic difference of our …
Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields. These manifolds generalize Inoue surfaces of type $S_m$. In this work it is shown that Oeljeklaus-Toma manifolds could not contain any compact complex submanifolds of dimension 2 (surfaces) except Inoue surfaces.
Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. We prove that Oeljeklaus-Toma manifolds contain no compact complex curves.
We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. In particular, we prove that Oeljeklaus-Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type $S^0$.
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.
The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of $U$ equal to the number of real embeddings of $K$. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper c…
We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any …
The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is two-fold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to Vaisman's question. We also point out a relation between lcK surfaces and generalized Kähler geometry in four-dimension and prove a new result concerning generalized Kähler structures on Hyperbolic Inoue surfaces. We conclude with a simple observation on a question of Brunella.
Oeljeklaus-Toma (OT) manifolds are higher dimensional analogues of Inoue-Bombieri surfaces and their construction is associated to a finite extension $K$ of $Q$ and a subgroup of units $U$. We characterize the existence of pluriclosed metrics (also known as strongly K\" ahler with torsion (SKT) metrics) on any OT manifold $X(K, U)$ purely in terms of number-theoretical conditions, yielding restrictions on the third Betti number $b_3$ and the Dolbeault cohomology group $H^{2,1}_{\overline{\partial}}$. Combined with the main result in [D20], these numerical conditions render explicit examples of…
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
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.
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 the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson mo…
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a blow-up operation for bi-Hermitian manifolds.
We give a direct proof of a cancellation formula raised in [7] on the level of differential forms. We also obtain more cancellation formulas for even dimensional Riemannian manifolds with a complex line bundle involved. Relations among these cancellation formulas are discussed.
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient condition is given for an almost $s-$contact metric manifold to be a generalized Kenmotsu manifold.We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we show t…
In this paper, invariant submanifolds of a generalized Kenmotsu manifold are studied. Necessary and sufficient conditions are given on a submanifold of a generalized Kenmotsu manifold to be an invariant submanifold.In this case, we investigate further properties of invariant submanifolds of \ a generalized Kenmotsu manifold. In addition, some theorems are given related to an invariant submanifold of a generalized Kenmotsu manifold.We consider semiparallel and 2-semiparallel invariant submanifolds of a generalized Kenmotsu manifold.
We define Aeppli and Bott-Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes $(A, d', d'')$, we show that the validity of $d'd''$-lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott-Chern cohomology groups of some bi-generalized Hermitian manifolds are given.
In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald manifold, we show that the notions of stretch and Landsberg curvatures are equivalent. Then we prove that on these manifolds, a Finsler metric is R-quadratic if and only if it is a stretch metric with vanishing E -curvature. Finally, we determine the flag curvature of generalized is…
The present paper deals with the study of generalized $ϕ$-recurrent generalized $(k,μ)$-contact metric manifolds with the existence of such notion by a proper example.
We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.
In this study, we investigate generalized quasi-Einstein structure for normal metric contact pair manifolds. Firstly, we deal with elementary properties and examine, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifold. Secondly, the generalized quasi-constant curvature of normal metric contact pair manifolds are studied and it is proven that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi typ…
To generalize the notion of recurrent manifold, there are various recurrent like conditions in the literature. In this paper we present a recurrent like structure, namely, \textit{super generalized recurrent manifold}, which generalizes both the hyper generalized recurrent manifold and weakly generalized recurrent manifold. The main object of the present paper is to study the geometric properties of super generalized recurrent manifold. Finally to ensure the existence of such structure we present a proper example by a suitable metric.
We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein-Weyl space. In particular, on the product $Z$ of any complex symplectic manifold $M$ and the sphere there exists a natural generalized complex structure, with respect to which $Z$ is the twistor space of …
The object of the present paper is to obtain the characterization of a warped product semi-Riemannian manifold with a special type of recurrent like structure, called super generalized recurrent. As consequence of this result we also find out the necessary and sufficient conditions for a warped product manifold to satisfy some other recurrent like structures such as weakly generalized recurrent manifold, hyper generalized recurrent manifold etc. Finally as a support of the main result, we present an example of warped product super generalized recurrent manifold.
We construct a natural generalized complex structure on the total space of any bundle endowed with a Chern connection and whose typical fibre is a homogeneous symplectic manifold. This extends known constructions of generalized complex structures on Lie groups and leads to natural examples of holomorphic maps between generalized complex manifolds.
In the paper we present results about generalized Berwald surfaces involving the intrinsic characterization, some topological obstructions for the base manifold and examples.
In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary generalized Killing spinors.
Tanno [6] provided an algebraic characterization in an almost Hermitian manifold to reduce to a space of constant holomorphic sectional curvature, which he later extended for the Sasakian manifolds as well. In this present paper, we generalize the same characterization in generalized $g.f.f-$manifolds.
We study differential forms and their higher-order generalizations by interpreting them as functions on map spaces. We get a series of approximations of "generalized manifolds" (i.e. of sheaves and stacks) somewhat akin to Taylor series.
We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was iniated by Pantilie, who shows that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space with an almost generalized complex structure. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperkähler manifolds using the generalized Bismut connection, introdu…
Generalized Roter type manifold is a generalization of conformally flat manifold as well as Roter type manifold, which gives rise the form of the curvature tensor in terms of algebraic combinations of the fundamental metric tensor and Ricci tensors upto level 2. The object of the present paper is to investigate the characterizations of a warped product manifold to be generalized Roter-type. We also present an example of a warped product manifold which is generalized Roter type but not Roter type, and also an example of a warped product manifold which is Roter type but not conformally flat. The…
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural non-associative algebra $\mathfrak{m}$ associated to the affine structure of the Stiefel manifold, computing explicitly the automorphism group $Aut(\mathfrak{m})$ and inducing this computation to the Isometry group.
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
In this paper, we establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, we prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, we give some applications of these generalized maximum principles.
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carathéodory pseudodistance on CR manifolds.
In this paper, we first get a criterion formula for whether a differential form is holomorphic with respect to the generalized complex structure induced by $ε$. Next, we get the local extensions of $\overline\partial$-closed forms on a smooth family of compact generalized Hermitian manifolds by using this criterion. Finally, as an application, we use this extension to get the invariance of the generalized Hodge number of the deformations of compact generalized Hermitian manifolds with $\partial\overline\partial$-lemma holds.
Let (g,b,J_+,J_-) be the bihermitian structure corresponding to a generalized Kaehler structure. We find natural integrability conditions under which db=0.
Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor
In this note, we consider submanifolds of a generalized Kähler manifold that are CR-submanifolds for the two associated Hermitian structures. Then, we establish the conditions for the induced, generalized F structure to be a CRFK structure. The results extend similar conditions which we obtained for hypersurfaces in an earlier paper.
We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkaehler, G_2 and Spin(7) manifolds.
Landweber and Stong prove that if a closed spin manifold $M$ admits a smooth $S^1$-action of odd type, then its signature $\mathrm{sign}(M)$ vanishes. In this paper, we extend the result to a torus action on a closed oriented manifold with generalized odd type.
Let M be a hyperkähler manifold. The S^2-family of complex structures compatible with the hyperkähler metric can be assembled into a single complex structure on Z=MxS^2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the sense of Hitchin. Specifically, we exhibit a natural S^2xS^2-family of generalized complex structures compatible with the hyperkähler metric, and assemble them into a single generalized complex structure on X=MxS^2xS^2. We call the resulting generalized complex manifold the general…
We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).
We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.
We give a local classification of generalized (kappa,mu)-Paracontact Metric Manifold which satisfies the condition xi(mu)=0. An example of such manifolds is presented.
