We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach-Lie groups.
This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anchor ranges.
This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. The main results of this paper generalize the works presented in [ChSt], [Nu] and [Gl].
We give an integrability criterion for a projective limit of Banach distributions on a Fréchet manifold which is a projective limit of Banach manifolds. This leads to a result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Fréchet setting. As consequence, we obtain a version of the third Lie theorem for a Fréchet-Lie group which is a submersive projective limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid.
To give a criterion for the integrability of Banach-Lie triple systems, we follow the construction of the period group of a Lie algebra and define the period group of a Lie triple system as an analogous concept. We show that a Lie triple system is integrable if and only if its period group is discrete. Along the way, we see how to turn the path and the loop space of a pointed symmetric space into pointed symmetric spaces.
Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the "Hilbert snake"
We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure.
We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details
This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Hermitian case of harmonic complex structures and the connections with balanced structures.
We survey some recent results and constructions of almost-Kähler manifolds whose curvature tensors have certain algebraic symmetries. This is an updated and corrected version of the (to be) published manuscript.
Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact almost Kähler manifolds with harmonic Weyl tensor and non-parallel semi-positive Ricci tensor. Stronger results are obtained in dimension 4.
The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and …
We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is…
Aguilar introduced isotropic almost complex structures $J_{δ, σ}$ on the tangent bundle of a Riemannian manifold $(M, g)$. In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics $g_{δ, σ}$ on $T M$ which are a generalization of the Sasaki metric. Moreover, the notion of a harmonic unit vector field is introduced with respect to these metrics like as the Sasaki metric and the necessary and sufficient conditions for a unit vector field to be a harmonic unit vector field are obtained.
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…
We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.
The classical theory of $G$-structures, which include almost-complex structures, explains the relationship between the curvature of compatible connections and integrability. This note is an effort to understand how the curvature of Riemannian metrics can obstruct the integrability of almost-complex structures. It is shown that certain special complex structures cannot coexist with non-flat constant curvature metrics, and a formal variational realization of these structures is provided. The approach followed here is direct, meaning that it bypasses the classical theory. The idea is to find obst…
If $W_+$ denotes the self dual part of the Weyl tensor of any Kähler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost Kähler 4-manifold with $S \ge 0$, this condition forces the Kähler property. A compact almost Kähler 4-manifold is already Kähler if it satisfies the conditions $| W_+ |^2 = S^2/6$ and $δW_+=0$ and also if it is Einstein and $| W_+|$ is constant. Some further results of this type are proved. An almost Hermitian 4-manifold $(M,g,J)$ with $\mathrm{supp} (W_+)=M$ is already Kähler if it satisfies the condition $| W_+ |^2 = …
We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kähler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-Kähler case
The natural bundle $π:E\to M$ of almost-complex structures is considered. The action of the pseudogroup of all diffeomorphisms of $M$ on the total space $E$ is investigated. A nontrivial 1-st order differential invariant of this action is constructed. It is proved that the Nijenhuise tensor of an almost-complex structures is equal to zero iff the constructed invariant for this structure is zero.
The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequenc…
This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not he…
An almost Golden Riemannian structure $(φ,g)$ on a manifold is given by a tensor field $φ$ of type (1,1) satisfying the Golden section relation $φ^{2}=φ+1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(φX,Y)=g(X,φY)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $φ$ and the integrability of the $G$-structure corresponding to $(φ,g)$.
In this paper we show that space of spatial polygons in semi riemann space gives a Kahler manifold. We describe the tangent space and almost complex structure which has many computational advantages.
We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. W…
We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.
We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.
We study $GL(2)$-structures on differential manifolds. The structures play a fundamental role in the geometric theory of ordinary differential equations. We prove that any $GL(2)$-structure on an even dimensional manifold give rise to a certain almost-complex structure on a bundle over the original manifold. Further, we exploit a natural notion of integrability for the $GL(2)$-structures, which is a counterpart of the self-duality for the 4-dimensional conformal structures. We relate the integrability of the $GL(2)$-structures to the integrability of the almost-complex structures. This allows …
We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the "transverse embeddability" of arbitrary compact complex manifolds into foliated algebraic varieties. In this work, we introduce a more general category of universal embedding spaces, and elucidate the geometric structure of the integrability locus characterizing integrable almost comp…
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.
