Paul-Andi Nagy
We consider complete nearly Kähler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly Kähler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly Kähler manifold to be a twistor space over a quaternionic Kähler manifold is given. This enables us to give classification results in 10-dimensions.
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We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a …
We study and classify almost complex totally geodesic submanifolds of the nearly Kaehler flag manifold $F_{1,2}(\mathbb C^3)$, and of its semi-Riemannian counterpart. We also develop a structural approach to the nearly Kaehler flag manifold $F_{1,2}(\mathbb C^3)$, expressing for example the curvature tensor in terms of the nearly Kaehler structure $J$ and the three canonical orthogonal complex structures.
In the present paper, we study the hemi-slant submanifolds of nearly Kaehler manifolds. We study the integrability of distributions involved in the definition of hemi-slant submanifolds. some results are worked out on totally umbilical hemi-slant submanifolds. we study the cohomology class for hemi-slant submanifolds of nearly Kaehler manifolds.
In this paper, we investigate Lagrangian submanifolds in the nearly Kaehler $S^3 \times S^3$. We construct a new example which is a at Lagrangian torus. We give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature in the nearly Kaehler $S^3\times S^3$.
We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.
We consider 6-dimensional strict nearly Kaehler manifolds acted on by a compact, cohomogeneity one automorphism group G. We classify the compact manifolds of this class up to G-diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G is simple.
In the present paper, we investigate geometric properties of Clairaut anti-invariant submersions whose total space is a nearly Kaehler manifold. We obtain condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. In the last, we introduce illustrative example.
Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G= SU_2 x SU_2, and M_reg its subset of regular points. We show that M_reg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kaehler structures and that a 1-parameter subfamily of such structures smoothly extend over a singular orbit of type S^3. This determines a new class of examples of nearly Kaehler structures on TS^3
Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in [17], it is interesting to find their existence. In this paper, we prove the existence of warped product semi-slant submanifolds of nearly Kaehler manifolds by a characterization. To this end we obtain an inequality for the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also discussed.
We study generic Riemannian submersions from nearly Kaehler manifolds onto Riemannian manifolds. We investigate conditions for the integrability of various distributions arising for generic Riemannian submersions and also obtain conditions for leaves to be totally geodesic foliations. We obtain conditions for a generic Riemannian submersion to be a totally geodesic map and also study generic Riemannian submersions with totally umbilical fibers. Finally, we derive conditions for generic Riemannian submersions to be harmonic map.
In some other context, the question was raised how many nearly Kähler structures exist on the sphere $§^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $λ= 12$ of the Laplacian acting on 2-forms. A similar result concerning nearly parallel $\G_2$-structures on the round sphere $§^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.
We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems.
We study bi-warped product submanifolds of nearly Kaehler manifolds which are the natural extension of warped products. We prove that every bi-warped product submanifold of the form $M=M_T\times_{f_1}\! M_\perp\times_{f_2}\! M_θ$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$\|h\|^2\geq 2p\|\nabla (\ln f_1)\|^2+4q\left(1+{\small \frac{10}{9}}\cot^2θ\right)\|\nabla(\ln f_2)\|^2,$$ where $p=\dim M_\perp$, $q=\frac{1}{2}\dim M_θ$, and $f_1,\,f_2$ are smooth positive functions on $M_T$. We also investigate the equality case of this inequality. Further, some applications …
We give an analog of triangle comparison for Kaehler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov-Hausdorff limits. We discuss tangent cones and singular Kaehler spaces.
The purpose of this paper is to establish a partial regularity theory on certain homogeneous complex Monge-Ampere equations. As consequences of this new theory, we prove the uniqueness of extremal Kaehler metrics and give an necessary condition for existence of extremal Kaehler metrics.
In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.
The basic class of the non-integrable almost complex manifolds with Norden metric is considered. Its curvature properties are studied. The isotropic Kaehler type of investigated manifolds is introduced and characterized geometrically.
Let $M$ be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.
Conformal Killing forms in Kaehler geometry
本文在Kähler几何中显式求解中间度数的共形Killing形式方程，完成紧致情形的分类，并首次给出非Hamiltonian 2-形式来源的实例，基于Calabi型流形，同时提供Hermitian Killing形式的结构与例子。
Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kaehler geometry.
There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any complete …
If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kaehler.
Page's Einstein metric on CP_2 # (-CP_2) is conformally related to an extremal Kaehler metric. Here we construct a family of conformally Kähler solutions of the Einstein-Maxwell equations that deforms the Page metric, while sweeping out the entire Kaehler cone of CP_2 # (-CP_2).The same method also yields analogous solutions on every Hirzebruch surface. This allows us to display infinitely many geometrically distinct families of solutions of the Einstein-Maxwell equations on the smooth 4-manifolds S^2 x S^2 and CP_2 # (-CP_2).
Two Kaehler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kaehler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected 2n-dimensional Riemannian Kaehler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we de…
Peng Wu recently announced a beautiful characterization of conformally Kaehler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det (W^+) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in (LeBrun 2015).
We consider compactification of type IIA supergravity on nearly Kaehler manifolds. These represent a simple class of SU(3) structure manifolds which includes S^6 and CP^3. We exhibit for the first time an explicit reduction ansatz in this context, obtaining an N=2 gauged supergravity in 4d with a single vector and hypermultiplet. We verify that supersymmetric solutions of the 4d theory lift to 10d solutions. Along the way, we discuss questions related to encountering both electric and magnetic charges in the 4d theory.
Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let $Δ_p$ and $D_p$ be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous results in the closed setting to show that (M,g,J) is Kaehler if and only if $Spec(Δ_p)=Spec(2D_p)$ for p=0,1. We also give a characterization of manifolds with constant sectional curvature or constant Ricci tensor (in the real setting) and manifolds of constant holomorphic sectional curvature (in the complex setting) in terms of spectral geometry.
In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension 4 presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. The present article, however, shows that techniques of $4$-dimensional symplectic geometry can be used to obtain all the major results of the previous pap…
In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real analytic pseudo-Riemannian manifold. In particular, for a pseudo-Riemannian homogeneous space, we define another complexification and a (complete) anti-Kaehler metric on the complexification. One of main purposes of this paper is to find the interesting relation between these two co…
In this paper, we initiate the study of $\p R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\p R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\p R$-warped products in the flat para-Kähler manifold with least codimension which satisfy the equality case of the inequality.
The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaehler manifold $\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\times_f N_\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\perp$ of $\tilde M$. In this article we survey recent results on warped product …
Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry
本文研究完备Ricci平坦ALF 4流形的形变，证明在一般衰减条件下形变保持厄米性并存在非平凡Killing场，且在温和附加条件下新度量属于已知分类族。
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds. As an application we have that a natural circle bundle over the Kuranishi moduli space of a Calabi-Yau threefold is a Lorentzian proper affine hypersphere.
Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topological obstructions to the existence of a geometrically 2-formal Kähler metric, at the level of the second cohomology group. A strong interaction with almost Kähler geometry is to be noted. In complex dimension 3, we list all the possible values of the second Betti number of a geome…
Let (M,I, ω, Ω) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ωand the Hermitian form ωwhich satisfies $dω=3λ\ReΩ, d\ImΩ=-2λω^2$, for a non-zero real constant λ. We develop an analogue of Kaehler relations on M, proving several useful identities for various intrinsic Laplacians on M. When M is compact, these identities bring powerful results about cohomology of M. We show that harmonic forms on M admit the Hodge decomposition, and prove that H^{p,q}(M)=0 unless p=q or (p=1, q=2) or (p=2, q=1).
In this paper we investigate the balanced condition (in the sense of Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on $C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced unless it is the flat metric. The second one shows that an Englis expansion of the Rawnsley's function associated to a LeBrun's metric always exists, while the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's metric is indeed the flat one.
We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the Kähler nullity of the almost complex structure. We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost Kähler manifolds, a class generalizing, to some algebraic extent, the …
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negative in the direction of a self-dual harmonic 2-form that is transverse to the zero section of the bundle of self-dual 2-forms. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely differe…
Let $(M,I)$ be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it admits a Hermitian form $ω$ such that $\nabla(ω)$ is totally antisymmetric, $\nabla$ being the Levi-Civita connection. We show that a nearly Kaehler metric on a given almost complex 6-manifold with non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly…
We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.
In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.
We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).
We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, RxG/H and R^2xG/H, where G/H is a compact nearly Kaehler six-dimensional homogeneous space, and the manifolds RxG/H and R^2xG/H carry G_2- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on RxG/H is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond…
The geometry that is defined by the scalars in couplings of Einstein-Maxwell theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler geometry. There are several equivalent definitions, the most elegant ones involve the symplectic duality group. The original construction used conformal symmetry, which immediately clarifies the symplectic structure and provides a way to make connections to quaternionic geometry and Sasakian manifolds.
We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in terms of root data. We also discuss when the Chern-Ricci form is a multiple of the symplectic form, and when compact quotients of these orbits are of Kaehler type.
Recently, we have shown that there do not exist the warped product semi-slant submanifolds of cosymplectic manifolds [10]. As nearly cosymplectic structure generalizes cosymplectic ones same as nearly Kaehler generalizes Kaehler structure in almost Hermitian setting. It is interesting that the warped product semi-slant submanifolds exist in nearly cosymplectic case while in case of cosymplectic do not exist. In the beginning, we prove some preparatory results and finally we obtain an inequality such as $\|h\|^2 \geq 4q\csc^2θ\{1+\frac{1}{9}\cos^2θ\}\|\nabla \ln f\|^2$ in terms of intrinsic and…
We review the general properties of target spaces of hypermultiplets, which are quaternionic-like manifolds, and discuss the relations between these manifolds and their symmetry generators. We explicitly construct a one-to-one map between conformal hypercomplex manifolds (i.e. those that have a closed homothetic Killing vector) and quaternionic manifolds of one quaternionic dimension less. An important role is played by 'ξ-transformations', relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds. In this map, the subclass of conformal hyper-Kae…
B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.
Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.
We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective…
We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.
We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1<r<b_1. As far as we know, these …
We consider an almost complex manifold with Norden metric (i. e. a metric with respect to which the almost complex structure is an anti-isometry). On such a manifold we study a linear connection preserving the almost complex structure and the metric and having a totally skew symmetric torsion tensor (i. e. a 3-form). We prove that if a non-Kaehler almost complex manifold with Norden metric admits such connection then the manifold is quasi-Kaehlerian (i. e. has non-integrable almost complex structure). We prove that this connection is unique, determine its form, and construct an example of it o…
We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kaehler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata conjecture for complete Kaehler manifold…
Generalized Kaehler geometry of instanton moduli spaces
本文证明了紧致偶广义Kaehler四流形上瞬子模空间的Hitchin广义Kaehler结构可通过广义Kaehler约化获得，其底层Courant algebroid约化实现了Donaldson三次$μ$-映射。
It is known that the almost-Kaehler anti-self-dual metrics on a given 4-manifold sweep out an open subset in the moduli space of anti-self-dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic 3-manifolds and self-dual harmonic 2-forms on associated 4-manifolds.
This is a survey of recent contributions to the area of special Kaehler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001.
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.
We give an explicit local classification of conformally equivalent but oppositely oriented Kaehler metrics on a 4-manifold which are toric with respect to a common 2-torus action. In the generic case, these structures have an intriguing local geometry depending on a quadratic polynomial and two arbitrary functions of one variable, these two functions being explicit degree 4 polynomials when the Kaehler metrics are extremal (in the sense of Calabi). One motivation for and application of this result is an explicit local description of Einstein 4-manifolds which are hermitian with respect to eith…
In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in this direction. In particular, we will discuss the progress recently obtained in understanding the metric structure of the infinite dimensional space of Kaehler potentials, and their applications to the problems mentioned above. We also will discuss some recent on Kaehler Ricci flo…
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