Flat nearly Kähler manifolds

正标量曲率4维流形的去奇异化

Desingularizing positive scalar curvature 4-manifolds

本文利用显式psc零配边及Lock–Viaclovsky的标量平坦Kähler ALE曲面，证明闭3维psc流形配边群平凡，并据此建立可扩大4维流形上奇异psc度量的不存在性及低正则渐近平坦4维流形的正质量定理。

Geometric structures on Lie groups with flat bi-invariant metric

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…

Transverse $J$-holomorphic curves in nearly Kähler $\mathbb{CP}^3$

$J$-holomorphic curves in nearly Kähler $\mathbb{CP}^3$ are related to minimal surfaces in $S^4$ as well as associative submanifolds in $Λ^2_-(S^4)$. We introduce the class of transverse $J$-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $S^4$ and construct moment-type maps from $\mathbb{CP}^3$ to relate them to the theory of $\mathrm{U}(1)$-invariant minimal surfaces on $S^4$.

On SNS-Riemannian connections in sub-Riemannian manifolds

The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate the geometric characteristics of the projective SNS-Riemannian connection, and obtain a necessary and sufficient condition for a nearly sub-Riemannian manifold being projectively flat.

Hypersurfaces of the homogeneous nearly Kähler $\mathbb{S}^6$ and $\mathbb{S}^3\times\mathbb{S}^3$ with anticommutative structure tensors

Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $ϕ$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $Aϕ+ϕA=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kähler manifold $\mathbb{S}^3\times\mathbb{S}^3$ does not admit a hypersurface that satisfies the condition $Aϕ+ϕA=0$.

Locally conformal parallel $G_2$ and $Spin(7)$ manifolds

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

Half-flat Structures and Special Holonomy

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…

On the behavior of quasi-local mass at the infinity along nearly round surfaces

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some geometric invariants of a family of nearly round surfaces which approach to infinity of the manifold.

Lifting SU(3)-structures to nearly parallel G_{2}-structures

Hitchin shows that half-flat SU(3)-structures on a 6-dimensional manifold M can be lifted to parallel G_{2}-structure on the product $M\times\mathbb{R}$. We show that Hitchin's approach can also be used to construct nearly parallel G_{2}-structures by lifting so-called nearly half-flat structures. These SU(3)-structures are described by pairs $(ω,ϕ)$ of stable 2- and 3-forms with $dϕ=λω^{2}$ for some $λ\in\mathbb{R}\setminus\{0\}$.

有限N热M理论中接触结构诱导的辛近半平SU(3)结构

Contact Structure-induced Symplectic Nearly Half-Flat $SU(3)$ Structure in Finite-$N$ Thermal ${\cal M}$-Theory

本文研究有限N热M理论的对偶G结构，证明其诱导的SU(3)结构属于辛近半平类，完成MQGP对偶的G结构分类，并给出有限温度弦论中弱磁场产生的全息机制。

Instantons on conical half-flat 6-manifolds

We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds redu…

Nearly hypo structures and compact Nearly Kähler 6-manifolds with conical singularities

We prove that any totally geodesic hypersurface $N^5$ of a 6-dimensional nearly Kähler manifold $M^6$ is a Sasaki-Einstein manifold, and so it has a hypo structure in the sense of \cite{ConS}. We show that any Sasaki-Einstein 5-manifold defines a nearly Kähler structure on the sin-cone $N^5\times\mathbb R$, and a compact nearly Kähler structure with conical singularities on $N^5\times [0,π]$ when $N^5$ is compact thus providing a link between Calabi-Yau structure on the cone $N^5\times [0,π]$ and the nearly Kähler structure on the sin-cone $N^5\times [0,π]$. We define the notion of {\it nearly…