Giovanni Russo
We review basic facts on the structure of nearly Kähler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly Kähler six-manifolds are Einstein is given by combining different known results. We finally rephrase the definition of nearly Kähler six-manifold in terms of a pair of partial differential equations.
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The aim of the paper is to investigate the rigidity and the deformability of pseudoholomorphic curves in the nearly K{ä}hler sphere $\mathbb{S}^6,$ among minimal surfaces in spheres. Under various assumptions we describe the moduli space of all noncongruent minimal surfaces $f\colon M\to\mathbb{S}^n$ that are isometric to a pseudoholomorphic curve in $\mathbb{S}^6.$ Moreover, we prove a Schur type theorem (see \cite[p. 36]{Chnew}) for minimal surfaces in spheres.
Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizing section, thus complementing earlier results on nearly-K{ä}hler almost complex structures.
We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define various quantities associated to such a hypersurface using the $\mathrm{G}_2$ structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the $\mathrm{G}_2$ structure of the ambient manifold, to be nearly K$\ddot{\text{a}}$hler. Then using the nearly $\mathrm{G}_2$ structure on the round sphere $S^7$, we prove that for a compact minimal hypersurface $M^6$ of constant scalar curvatur…
We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^1$. We partition the class of such nearly-Ka}hler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on the 2-dimen…
The subject of this paper is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)Kähler structures on Lie groups $G \times G$ where $G$ is three-dimensional and simple.
We consider nearly Kähler 6-manifolds with effective 2-torus symmetry. The multi-moment map for the $T^2$-action becomes an eigenfunction of the Laplace operator. At regular values, we prove the $T^2$-action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. An inverse construction is given locally producing nearly Kähler six-manifolds from three-dimensional data. This is illustrated for structures on the Heisenberg group.
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
Hypersurfaces of six-dimensional nearly Kähler manifolds
该研究在六维齐次近Kähler流形框架下，证明$\u211d^6$是唯一存在常截面曲率超曲面的外围空间，并刻画了该类流形中Sasakian等四类特殊超曲面，推进了近Kähler几何的子流形研究。
We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.
In this paper we find examples of slant surfaces in the nearly Kahler six sphere. First, we characterize two-dimensional small and great spheres which are slant. Their description is given in terms of the associative 3-form in $\Im \OO .$ Later on, we classify the slant surfaces of $S^6$ which are orbits of maximal torus in $G_2.$ We show that these orbits are flat tori which are linearly full in $S^5\subset S^6$ and that their slant angle is between $\arccos \frac{1}{3}$ and $\fracπ{2}.$ Among them we find one parameter family of minimal orbits.
We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non Kähler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic Kähler manifolds $Q\sp{4n}$ are minimal. Moreover, we will prove that any Lagrangian submanifold $L$ in a nearly Kähler manifold $M$ splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of $M$ and the second factor is Lagrangian in the Kähler part of $M$. Using this splitting theorem we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight and ten.
Isometry groups of nearly Kähler manifolds
本文用替代且较少代数的方法，显式给出了六维齐次近凯勒流形的等距群。
Sharp scalar and Ricci curvature pinching theorems for Lagrangian submanifolds in the homogeneous nearly Kähler Six-sphere
本文针对齐性近凯勒六球面中的闭拉格朗日子流形，建立新的积分不等式，并证明数量曲率与里奇曲率的锐利夹逼定理，刻画全测地子流形与Berger球面的特征。
A Quantitative Reeb-Form Obstruction for Real Hypersurfaces of Strict Nearly Kähler Six-Manifolds
本文证明六维严格近凯勒流形的实超曲面上Reeb一次形式的外微分逐点非零，并由此得出Yano联络的Ricci张量处处不对称，排除了满足dη=0的诱导几乎切触类。
Firstly we give a condition to split off the K"ahler factor from a nearly pseudo-K"ahler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly K"ahler manifolds from twistor spaces to negatively curved quaternionic K"ahler manifolds and para-quaternionic K"ahler manifolds. The class of nearly pseudo-K"ahler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-K"…
The structure of nearly Kähler manifolds was studied by Gray in several papers. More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a strict and complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over quaternionic K{ähler manifolds and six-dimensional nearly Kähler manifolds, where the homogeneous nearly Kähler factors are also 3-symmetric spaces. In the present paper, using the lists of $3$-symmetric spaces given by Wolf & Gray, we display the exhaustive list of irreducibl…
We give a systematic way to construct almost conjugate pairs of finite subgroups of $Spin(2n+1)$ and $Pin(n)$ for $n\in \mathbb{N}$ sufficiently large. As a geometric application, we give an infinite family of pairs $M_1^{d_n}$ and $M_2^{d_n}$ of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions $d_n>6$. We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.
We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly Kähler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bu…
We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are $S^3 \times S^3$, the complex projective space $\CM P^3$, the flag manifold $\mathbb F^3$ and the sphere $S^6$. We develop, about each of these spaces, a distinct aspect of nearly Kähler geometry and make in the same time a sharp description of its specific homogeneous structure.
Let L\subset V=\bR^{k,l} be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η\in Λ^3L\subset Λ^3V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphism J with J^2=\e Id, then the Lie group {\cal L}(η) is endowed with a left-invariant nearly Kähler structure if \e =-1 and with a left-invariant nearly para-Kähler structure if \e =+1. This construction exhausts all complete simply connected flat nearly (para-)Kähler manifolds. If…
Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature
本文研究四维伪脐双调和子流形，通过引入λ-双极小子流形，证明六维欧氏空间中此类子流形必为极小，部分回答了Chen猜想。
It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of…
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 …
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…
About generalized complex structures on $\mathbb S^6$
本文研究六维球面 $\mathbb S^6$ 上广义复结构的存在性，通过广义切丛和 Dorfman 括号定义可积性，证明无法由其八元数乘积继承的近凯勒结构直接诱导出广义复结构。
We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly Kähler manifold. We show that in these cases the $G_{2}$-structures are determined by their torsion components up to a phase factor. We then study the modified Laplacian coflow $\frac{dψ}{dt}=Δ_{ψ}ψ+2d\left( \left( C-Tr T\right) φ\right) $ of these $G_{2}$-structures, where $φ$ and $ψ$ are the fundamental $3$-form and $4 $-form which define the $G_{2}$-structure and $Δ_{ψ}$ is the Hodge Laplacian associated with the $G_{2}$-structure. Th…
We consider G-equivariant dimensional reduction of Yang-Mills theory with torsion on manifolds of the form MxG/H where M is a smooth manifold, and G/H is a compact six-dimensional homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures which includes a nearly Kahler structure. We establish an equivalence between G-equivariant pseudo-holomorphic vector bundles on MxG/H and new quiver bundles on M associated to the double of a quiver Q, determined by the SU(3)-structure, with relations ensuring the absence of oriented cycles in Q. When M=R^2, …
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