Vector bundles with infinitely many souls

Metrics with Nonnegative Curvature on S^2xR^4

We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metric on S^2xR^4 (so that Wilking's space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.

复线丛上无限多个非坍缩的稳态里奇溶剂

Infinitely many non-collapsed steady Ricci solitons on complex line bundles

研究在复射影空间上的复线丛O(k)上构造无限多个非收缩的里奇溶剂，并证明其具有渐近锥形的里奇平坦度量。

Space of nonnegatively curved metrics and pseudoisotopies

Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the space of complete nonnegatively curved metrics has infinite higher homotopy groups.

Simply and tangentially homotopy equivalent but non-homeomorphic homogeneous manifolds

For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. These manifolds are the fi…

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an inf…

切丛不是双商向量丛的双商流形示例

Examples of biquotients whose tangent bundle is not a biquotient vector bundle

本文研究双商流形切丛何时为双商向量丛，构造了无穷多族切丛不是双商向量丛的双商流形，并给出同一流形两种双商呈现下切丛性质不同的无穷多族例子。

带边界的佐尔曼体

Zoll manifolds with boundary

本文研究了带边界的佐尔曼 manifold：紧致黎曼流形，其边界光滑，且每条从边界正交出发的测地线都正交返回且不与之相切。研究发现所有自由边界测地线都是嵌入的，具有相同长度，边界至多有两个连通分支。若有两个分支，则流形是区间与闭合流形的积；当边界连通时，流形是闭合嵌入子流形的管状邻域，其补集与边界上的半开圆柱面同胚。所有自由边界测地线都是能量函数的极值点，具有相同的莫尔斯指数，等于每个测地线中点处唯一焦点点的乘数。边界到灵魂的投影是二重覆盖或光滑球丛。应用包括对带边界的佐尔表面和三维佐尔曼体的完全分类。

Curvature and symmetry of Milnor spheres

In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negati…

具有无穷多种球上纤维化结构的流形

Manifold with infinitely many fibrations over the sphere

本文证明 $S^2\ imes S^3$ 在 $S^2$ 上容许无穷多种纤维丛结构，且每个透镜空间 $L(p,1)$ 都可作为其纤维。

On non-negatively curved metrics on open five-dimensional manifolds

Let $V^n$ be an open manifold of non-negative sectional curvature with a soul $Σ$ of co-dimension two. The universal cover $\tilde N$ of the unit normal bundle $N$ of the soul in such a manifold is isometric to the direct product $M^{n-2}\times R$. In the study of the metric structure of $V^n$ an important role plays the vector field $X$ which belongs to the projection of the vertical planes distribution of the Riemannian submersion $π:V\toΣ$ on the factor $M$ in this metric splitting $\tilde N=M\times R$. The case $n=4$ was considered in [GT] where the authors prove that $X$ is a Killing vect…

Biquotient vector bundles with no inverse

In previous work, the second author and others have found conditions on a homogeneous space $G/H$ which imply that, up to stabilization, all vector bundles over $G/H$ admit Riemannian metrics of non-negative sectional curvature. One important ingredient of their approach is Segal's result that the set of vector bundles of the form $G\times_H V$ for a representation $V$ of $H$ contains inverses within the class. We show that this approach cannot work for biquotients $G/\!\!/ H$, where we consider vector bundles of the form $G\times_{H} V$. We call such vector bundles biquotient bundles. Specifi…

Determinat Bundles and Geometric Quantization Of Vortex Moduli Spaces ON Compact Kahler Surfaces

In this paper we first show that on projective manifolds (M, ω), there are holomorphic determinant bundles (in the sense of Knusden-Mumford used by Bismut, Gillet, Soule) which play the role of the geometric quantum bundle, namely one for each input data of a Hermitian holomorphic line bundle L of non-trivial Chern class on a compact Kahler manifold Z (with Todd genus non-zero) and a choice of a geometric quantization of (M, ω). Next we further study the generalization of the vortex equations on Kahler 4-manifold which has been studied earlier by Bradlow. We show that when the Kahler 4-manifol…

嵌入 $\mathbb{S}^4$ 中的极小 $S^1$-丛

Embedded minimal $S^1$-bundles in $\mathbb{S}^4$

利用等变极小极大理论和悬垂（加权）Hopf作用，构造了 $\mathbb{S}^4$ 中无穷多个不同不可约拓扑类型的极小超曲面，并证明任何奇数亏格的闭可定向曲面上的 $S^1$ 丛可极小嵌入。

来自渐近锥形Calabi-Yau流形上$T^2$-丛的完备非紧$\u200b\operatorname{Spin}(7)$-流形

Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds

本研究提出在渐近锥形Calabi-Yau流形的主T²丛上构造完备非紧Spin(7)流形的新方法，得到渐近T²纤维化锥形的新型无穷远几何，并构造出无穷多微分同胚型及首个完备环形Spin(7)流形实例。

Feuilletage canonique sur le fibré de Weil

Let be M a smooth manifold, A a local algebra and M^{A} a manifold of infinitely near points on M of kind A. We build the canonical foliation on M^{A} et we show that the canonical foliation on the tangent bundle TM is the foliation defined by his canonical field.

Highly connected 7-manifolds, the linking form and non-negative curvature

In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert fibration with generic fibre $S^3$ and, in particular, has the cohomology ring of an $S^3$-bundle over $S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $S^3$-bundle over $S^4$, the first time that any such spaces hav…

Fibrations with constant scalar curvature Kahler metrics and the CM-line bundle

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature Kahler metric on the total space X. The condition involves the CM-line bundle--a certain natural line bundle on B--which is proved to be nef. Knowing this, the condition is then implied by c_1(B)<0. This provides infinitely many Kahler manifolds of constant scalar curvatu…

On the space of connections having non-trivial twisted harmonic spinors

We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.

大体积流形上带正标量曲率的圆丛

Circle bundles with PSC over large manifolds

本研究利用辛几何技术构造了无穷多类m≥4维宏观大体积流形的正标量曲率圆丛实例，其全空间宏观维度不超过⌈m/2⌉+1，并全维度解答了Gromov关于可扩流形上此类圆丛存在性的问题。

Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S^2-bundle over H^4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we …

Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Mañé critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. We show that, under a suitable criticality assumption on a given Randers metric…

全旗流形上的变形厄米-杨-米尔斯方程

Deformed Hermitian-Yang-Mills equation on the manifold of full flags

研究全旗流形上变形厄米-杨-米尔斯方程，构建小半径高阶不可约及大半径无穷多不同丛的dHYM联络示例，研究超临界外解，证明任意角度解存在并排除部分稳定性条件。

Einstein metrics on connected sums of $S^2\times S^3$

The aim of this paper is to construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of $S^2\times S^3$. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A Kähler--Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi--Boyer--Galicki method.

自由环面作用与扭曲悬挂

Free torus actions and twisted suspensions

本文将两流形连通和上的主圆丛全空间表示为各加项上的圆丛全空间，以此为球面乘积连通和上的自由圆与环面作用提供存在充分条件，并对具自由余齐次四环面作用的闭单连通流形进行拓扑分类，推得无限多具正 Ricci 曲率与等距环面作用的流形。

$T$-Dual solutions and infinitesimal moduli of the $G_2$-Strominger system

We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dime…

$\mathbb{C}^3$ 的 Engel CR 子流形

Engel CR submanifolds of $\mathbb{C}^3$

本文给出了3维流形上$\mathbb{S}^1$丛浸入或嵌入$\mathbb{C}^3$且复切定义Engel结构的充分条件，构造了首批复切为Engel的紧子流形实例，回答了Y. Eliashberg的问题。

The spectrum of an operator associated with $G_{2}-$instantons with $1-$dimensional singularities and Hermitian Yang-Mills connections with isolated singularities

This is the first step in an attempt at a deformation theory for $G_{2}-$instantons with $1-$dimensional conic singularities. Under a set of model data, the linearization yields a self-adjoint first order elliptic operator $P$ on a certain bundle over $\mathbb{S}^{5}$. As a dimension reduction, the operator $P$ also arises from Hermitian Yang-Mills connections with isolated conic singularities on a Calabi-Yau $3$-fold. Using the Quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$ (denoted by $Spec P$). We show that $SpecP$ consists o…