In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost Kähler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost Kähler geometry and explore the generalized Monge-Ampère equation on almost Kähler manifolds. We establish a uniqueness up to the addition of a constant and local existence theorem for this equation. At last, we find an elliptical system for $\mathcal{D}^+_J$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.
Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
In this note we prove the following result: There is a positive constant $ε(n,Λ)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $Λ$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -ε(n,Λ)$, then $M^n$ is diffeomorphic to the product $M_1\times ... \times M_k$, where each $M_i$ is either a complex projective space or an irreducible K$\ddot{a}$hler-Hermitian symmetric space of rank $\geq 2$. This resolves a conjecture of F. Fang under the additional upper bound restrictions on section…
In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:C \rightarrow X$.
We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results to almost hyperhermitian geometry. Thus, we show that the exterior derivatives of the three K{ä}hler forms of an almost hyperhermitian manifold are sufficient to determine the three covariant derivatives of such forms, i.e., the three mentioned exterior derivatives determine the …
The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galicki and Piccinni in \emph{$3-$Sasakian geometry, nilpotents orbits, and exeptional quotients}. We construct explicitly our new examples as quaternion-K$\ddot{\mathrm{a}}$hler reductions of the quaternion K$\ddot{\mathrm{a}}$hler Grassmannian $Gr_4(\mathbb{R}^8)$ by an isometric act…
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…
The space ${\mathcal A}$ of almost complex structures on a closed manifold $M$ is studied. A natural parametrization of the space ${\mathcal A}$ is defined. It is shown, that ${\mathcal A}$ is a infinite dimensional complex weak Pseudo-Riemannian manifold. A curvature of the space ${\mathcal A}$ is found. The space ${\mathcal A}_ω$ of associated almost complex structures on a symplectic manifold $M,ω$ and space ${\mathcal AO}$ of orthogonal almost complex structures on a Riemannian manifold $M,g_0$ are considered in more detail.
Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\mathcal{F}$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\mathcal{F}$. The corresponding Lie algebra $\mathfrak{g}$ must then belong to one of 20 families $\mathfrak{g}_1,\dots,\mathfrak{g}_{20}$ according to S. Gudmundsson and M. Svensson. We classify such structures $J$ which are almost Kähler $(\mathcal{A}\mathcal{K})$, integrable $(\mathcal{I})$ or Kähler $(\mathcal{K})$. Hereby, we construct 16 multi-dimensional almost Kähler fami…
We study 4-dimensional Riemannian manifolds equipped with a minimal and conformal foliation $\mathcal F$ of codimension 2. We prove that the two adapted almost Hermitian structures $J_1$ and $J_2$ are both cosymplectic if and only if $\mathcal F$ is Riemannian and its horizontal distribution $\mathcal H$ is integrable.
To give an almost quaternionic structure on a 4n-manifold $M$ is equivalent to give its bundle of twistors $Z(Q)\longrightarrow M$. When $Q$ is invariant under a torsion free connection, $Z(Q) $ can be provided with an almost complex structure $ \mathbb J $. In the case $ n = 1 $ Atiyah, Hitchin and Singer have related the integrability of $ \mathbb J $ to the geometry of $ (M, Q) $. For $ n> 1 $ Salamon showed that the almost complex structure $ \mathbb J $ on $ Z (Q) $ is always integrable. The purpose of this article is to extend these results to the generalized complex geometry. We begin b…
An $f$-structure on a manifold $M$ is an endomorphism field $ϕ$ satisfying $ϕ^3+ϕ=0$. We call an $f$-structure {\em regular} if the distribution $T=\kerϕ$ is involutive and regular, in the sense of Palais. We show that when a regular $f$-structure on a compact manifold $M$ is an almost $§$-structure, as defined by Duggal, Ianus, and Pastore, it determines a torus fibration of $M$ over a symplectic manifold. When $\rank T = 1$, this result reduces to the Boothby-Wang theorem. Unlike similar results due to Blair-Ludden-Yano and Soare, we do not assume that the $f$-structure is normal. We also sh…
We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjecture: $\dim(H^*(M;\mathbb{Q})) \geq 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.
Donaldon constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $Σ$ with $\textrm{genus}(Σ) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(Σ)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $Σ$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bun…
We study $\mathcal D$-homothetic deformations of almost $α$-Kenmotsu structures. We characterize almost contact metric manifolds which are $CR$-integrable almost $α$-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathcal D$-homothetic deformations. If the canonical connection associated to the structure $(φ,ξ,η,g)$ has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator $h'$ defined by $2αh'=({\mathcal L}_ξφ)\circφ$. In particular, the manifold is locall…
We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian st…
The space $\mathcal{Z}$ of leftinvariant orthogonal almost complex structures, keeping the orientation, on 6-dimensional Lie groups is researched. To get explicit view of this space elements the isomorphism of $\mathcal{Z}$ and $\mathbb{C}P^3$ is used. The explicit formula for arbitrary leftinvariant orthogonal almost complex structure on 6-dimensional Lie group as composition of rotations is found.
We study a cohomology theory $H^{\bullet}_φ$, called the $\mathcal L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that $H^k_φ \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that $H^k_φ$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_φ$. The $\mathcal L_B$-cohomology also satisfies a Poincaré duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation $\mathcal L_B$, and uses both Hodge…
In this paper we describe the oriented Riemannian four-manifolds $M$ for which the Atiyah-Hitchin-Singer or Eells-Salamon almost complex structure on the twistor space ${\mathcal Z}$ of $M$ determines a harmonic map from ${\mathcal Z}$ into its twistor space.
In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly Kähler manifolds. Besides its almost complex structure $J$ it also admits a canonical almost product structure $P$. Along a $3$-dimensional CR submanifold the tangent space of $\mathbb S^3\times\mathbb S^3$ can be naturally split as the orthogonal sum of three $2$-dimensional vector bundles $\mathcal D_1$, $\mathcal D_2$ and $\mat…
Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $κ_p \in \mathbf{R}$ such that $\mathcal{J}_v = κ_p \textit{ Id}_{v^\perp}$ for each unit vector $v \in T_pM$. If all points are isotropic, then $M$ is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. …
In this paper, we prove a pseudolocality-type theorem for $\mathcal L$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the $\mathcal L$-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L.~Chen.
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}£_ξφ$ and $\ell := R(\cdot,ξ)ξ$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $ξ$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(\mathcal{H}(M), J, θ)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,φ,ξ,η,g)$ is Sasakian if and only if the corresponding nonde…
In this short note it is established that there does not exist Ricci soliton with the Reeb potential vector field in an almost Kenmotsu manifold (briefly, $\mathcal{AKM}$).
The vector space of the tensors $\mathcal F$ of type (0,3) having the same symmetries as the covariant derivative of the fundamental form of an almost contact metric manifold is considered. A scheme of decomposition of $\mathcal F$ into orthogonal components which are invariant under the action of $U(n)\times 1$ is given. Using this decomposition there are found 12 natural basic classes of almost contact metric manifolds. The classes of cosymplectic, $α$-Sasakian, $α$-Kenmotsu, etc. manifolds fit nicely to these considerations. On the other hand, many new interesting classes of almost contact …
We present a compared analysis of some properties of indefinite almost $\mathcal{S}$-manifolds and indefinite $\mathcal{S}$-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and $ϕ$-sectional curvature of indefinite almost $\mathcal{S}$-manifolds and state an expression of the curvature tensor field for the indefinite $\mathcal{S}$-space forms. We analyse the sectional curvature of indefinite $\mathcal{S}$-manifold in which the number of the spacelike characteristic vector fields is equal to that of …
We compute the condition of minimality of a G-structure for the Gray-Hervella class $\mathcal{W}_4$ of almost hermitian manifolds and $\mathcal{C}_5$ class of almost contact metric structures. We also consider $\mathcal{C}_4$ class by comparison with the Grey-Hervella class $\mathcal{W}_4$. The common feature is the existence of the Lee form $θ$ representing these structures. We show that these classes contain minimal G-structures. Here, minimality means minimality of a $G$-structure inside oriented orthonormal frame bundle $SO(M)$ of a Riemannian manifold $M$.
In an earlier work, we constructed the almost strict Morse $n$-category $\mathcal X$ which extends Cohen $\&$ Jones $\&$ Segal's flow category. In this article, we define two other almost strict $n$-categories $\mathcal V$ and $\mathcal W$ where $\mathcal V$ is based on homomorphisms between real vector spaces and $\mathcal W$ consists of tuples of positive integers. The Morse index and the dimension of the Morse moduli spaces give rise to almost strict $n$-category functors $\mathcal F : \mathcal X \to \mathcal V$ and $\mathcal G : \mathcal X \to \mathcal W$.
We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex struc…
We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost Kähler manifolds $(M,J,ω)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $ψ\in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $ω^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provi…
The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kähler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined me…
An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost K…
We determine the Hamiltonian vector field on an odd dimensional manifold endowed with almost cosymplectic structure. This is a generalization of the corresponding Hamiltonian vector field on manifolds with almost transitive contact structures, which extends the contact Hamiltonian systems. Applications are presented to the equations of motion on a particular five-dimensional manifold, the extended Siegel-Jacobi upper-half plane $\tilde{\mathcal{X}}^J_1$. The $\tilde{\mathcal{X}}^J_1$ manifold is endowed with a generalized transitive almost cosymplectic structure, an almost cosymplectic structu…
A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem and we describe almost hermitian $\mathrm{G}_1$-manifolds with parallel torsion form. In particular, among them there are only two types of $\mathcal{W}_3$-manifolds with a non-abelian holonomy group, namely twistor spaces of 4-dimensional self-dual Einstein manifolds and the invar…
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 $M$.
In this paper, we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. A…
If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some $x\in X$ whose stabilizer $G_x\leq G$ satisfies $[G:G_x]\leq C$. We say that $X$ has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if $X$ is compact and possibly with boundary and has no odd cohomology then $(X,Diff(X))$ has the almost fixed point property. Combining this with a result of Petrie an…
