Diffeomorphism type of 6-dimensional cohomogeneity one manifolds

On the homology of low dimensional cohomogeneity one manifolds

In this paper we give a characterization of the possible homology groups that can occur for compact simply connected cohomogeneity one manifolds in dimensions seven and lower.

Cohomogeneity one manifolds with a small family of invariant metrics

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.

Seven dimensional cohomogeneity one manifolds with nonnegative curvature

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family of candidates

Cohomogeneity One Manifolds and Homogeneous Spaces of Positive Scalar Curvature

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

Classification of Cohomogeneity One Manifolds in Low Dimensions

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

Cohomogeneity one manifolds with singly generated rational cohomology II

We classify cohomogeneity one actions on smooth, simply connected, closed manifolds with the rational cohomology of a sphere. In particular, we show that such a manifold is diffeomorphic to a sphere, a Brieskorn variety, the Wu manifold $SU(3)/SO(3)$, one of two homogeneous spaces of the form $\mathbf{G}_2/SU(2)$,or a member of a particular four-parameter family of $7$-manifolds.

G2 manifolds of cohomogeneity two

This paper has been withdrawn by the author, due to errors in Groebner basis calculations in the cases of five and six dimensional groups.

Cohomogeneity one manifolds with quasipositive curvature

In this paper we give a classification of cohomogeneity one manifolds admitting an invariant metric with quasipositive sectional curvature except for two $7$-dimensional families. The main result carries over almost verbatim from the classification results in positive curvature carried out by Verdiani and Grove, Wilking and Ziller. Three main tools used in the positively curved case that we generalized to quasipositively curved cohomogeneity one manifolds are Wilking's Chain Theorem, the classification of positively curved fixed point homogeneous manifolds by Grove and Searle and the Rank Lemm…

Nonnegative Curvature on Low Dimensional Cohomogeneity One Manifolds

This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking.

Eight-dimensional SU(3)-manifolds of cohomogeneity one

In this paper, we classify 8-dimensional manifolds M admitting an SU(3) action of cohomogeneity one such that (i) M is simply connected and the orbit space M/G is isomorphic to [0,1], and (ii) M/G=S^1 and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to 8-dimensional quaternion-Kahler spaces.

On the geometry of cohomogeneity one manifolds with positive curvature

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics.

Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among the 7-dimensional Eschenburg spaces and 13-dimensional Bazaikin spaces and one isolated 7-dimensional Berger space with this property. In addition to these examples, it turns out that only one isolated 7-manifold, and two infinite 7-dimensional families, potentially admit invaria…

Harmonic self-maps of cohomogeneity one manifolds

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear solutions to non-linear singular boundary value problems.

闭$4$流形上的余齐一爱因斯坦度量

Cohomogeneity-one Einstein metrics on closed $4$-manifolds

分类了等距群维数至少为3的闭4流形上的爱因斯坦度量，证明闭4流形上的余齐一爱因斯坦度量局部对称或与$\mathbf{CP}^2 \sharp \overline{\mathbf{CP}}^2$上的Page度量位似。

New Examples of Obstructions to Non-Negative Sectional Curvatures in Cohomogeneity One Manifolds

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.

关于余维数为一的调和映射的初值问题

The Initial Value Problem for Harmonic maps of Cohomogeneity One manifolds

本文研究了余维数为一的调和映射的初值问题，证明了在奇点轨道邻域内调和映射的局部存在性，并探讨了正则-奇点系统的一阶理论。

余齐性三中的正曲率与有理椭圆性

Positive curvature and rational ellipticity in cohomogeneity three

本文证明了商空间无边界的闭、单连通、正曲率余齐性三流形是有理椭圆的，推广了齐性、余齐性一及余齐性二流形的相关结果。

Six-dimensional nearly Kaehler manifolds of cohomogeneity one (II)

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

上同调一维流形上的初值问题，I

Initial value problems on cohomogeneity one manifolds, I

该研究聚焦上同调一维流形奇异轨道附近的几何方程初值问题，证明指定Ricci曲率、求解Einstein及孤立子方程时存在有限常数自由度下的唯一解，第I部分采用简化假设，一般情形留待第II部分解决。

Delaunay-type hypersurfaces in cohomogeneity one manifolds

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres.

低维等余一流形上的正中间 Ricci 曲率

Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

研究闭低维等余一流形上具有正中间 Ricci 曲率的不变度量的存在性，对特定作用构造了正第4中间 Ricci 曲率度量并证明不能有正第3中间 Ricci 曲率度量，同时建立了类似对称性障碍。

关于具有余齐次二对称性的 $G_2$ 流形

On $G_2$ manifolds with cohomogeneity two symmetry

本文研究了具有余齐次二对称性的 $G_2$ 流形，给出了局部刻画及余齐次一校准子流形的几何与奇性分析，并在特定流形中发现了新的完备 $\mathbb{T}^2$-不变结合子流形族。

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \rank H)\le 5$. Moreover, we prove that the Euler characteristic of a compact Riemannian manifold $M^{4l+4}$ or $M^{4l+2}$ with positive sectional curvature is positive if $M$ admits an effective isometric action of a torus $T^l$, i.e., if the symmetry rank of $M$ is $\ge l$.

上同调一维黎曼流形的Ambrose-Singer定理

The Ambrose-Singer Theorem for cohomogeneity one Riemannian manifolds

本文研究上同调一维黎曼流形，借鉴齐性空间Ambrose-Singer定理思路，通过满足协变方程的线性联络刻画主轨道为超曲面的等距作用，以此描述等距上同调一维叶状结构，并给出欧氏空间与实双曲空间中的具体实例。

关于余齐性一流形上调和自映射的等变稳定性

On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds

本文研究余齐性一流形上调和自映射的等变稳定性，分析相应Jacobi方程，并在球面、特殊正交群及SU(3)情形显式求解，证明球面上特定作用下恒等映射等变稳定。

Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus $T^{n-2}$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Cohomogeneity one Kahler and Kahler-Einstein manifolds with one singular orbit, I

Let $M$ be a cohomogeneity one manifold of a compact semisimple Lie group $G$ with one singular orbit $S_0 = G/H$. Then $M$ is $G$- diffeomorphic to the total space $G \times_H V$ of the homogeneous vector bundle over $S_0$ defined by a sphere transitive representation of $G$ in a vector space $V$. We describe all such manifolds $M$ which admit an invariant Kahler structure of standard type. This means that the restriction $μ: S = Gx = G/L \rightarrow F = G/K$ of the moment map of $M$ to a regular orbit $S = G/L$ is a holomorphic map of $S$ with the induced CR structure onto a flag manifold $…

Smoothness Conditions in Cohomogeneity manifolds

In this paper we discuss the smoothness conditions for metrics on a cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose generic orbits are hypersurfaces. Along these hypersurfaces one describes the metrics in terms of a collection of functions defined along a geodesic normal to the hypersurfaces. In a neighborhood of a lower dimensional orbit the functions must satisfy certain smoothness conditions for the metric to extend smoothly. In general these can be quite complicated. We present a method that makes it straightforward to compute them, and illustrate it in several e…

Nearly-Kahler 6-Manifolds of Cohomogeneity Two: Local Theory

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…

关于余齐一六维流形上平衡度量的存在性

On the existence of balanced metrics on six-manifolds of cohomogeneity one

本文证明了单连通六维流形上余齐一作用下不变的平衡非凯勒SU(3)结构不存在，为Hull-Strominger系统提供了理论限制。

Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II

F. Podestà and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part of this paper, we gave a combinatoric description of the standard non compact $G$-manifolds as the total space $M_φ$ of the homogeneous vector bundle $M = G\times_H V \to S_0 =G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-Eins…

The Dirichlet Problem for Einstein Metrics on Cohomogeneity One Manifolds

Let $G/H$ be a compact homogeneous space, and let $\hat{g}_0$ and $\hat{g}_1$ be $G$-invariant Riemannian metrics on $G/H$. We consider the problem of finding a $G$-invariant Einstein metric $g$ on the manifold $G/H\times [0,1]$ subject to the constraint that $g$ restricted to $G/H\times \{0\}$ and $G/H\times \{1\}$ coincides with $\hat{g}_0$ and $\hat{g}_1$, respectively. By assuming that the isotropy representation of $G/H$ consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.

Cohomogeneity one topological manifolds revisited

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions $5$, $6$, and $7$ and obtain topological characterizations of these spaces. In these dimensions, these manifolds are homeomorphic to smooth manifolds.

Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres

In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case where both are equal to two, where existence of non-negatively curved metrics is known.

具有余二维环面作用的单连通流形上的正里奇曲率

Positive Ricci curvature on simply-connected manifolds with cohomogeneity-two torus actions

本文针对余二维环面作用的闭单连通流形，通过替代证明，在无限子族上构造了正里奇曲率度量，并推广到所有允许该作用的5维和6维流形。

On cohomogeneity one Hermitian non-Kähler metrics

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb C\mathbb P^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman, Gauduchon. Furthermore, we construct new examples …

On Ricci solitons of cohomogeneity one

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kahler-Einstein manifolds or over coadjoint orbits of a compact connected semisimple Lie group.

Cohomogeneity one actions on symmetric spaces of noncompact type

An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As a consequence, we find many new examples of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type. We apply our conceptual approach to derive explicit classifications of cohomogeneity one actions on some symmetric spaces.

Cohomogeneity One Coassociative Submanifolds in the Bundle of Anti-self-dual 2-forms over the 4-sphere

Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $Λ^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of $Λ^{2}_{-} S^{4}$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical…

Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.

Cohomogeneity One Alexandrov Spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contrast to the 1- and 2-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of $RP^2$, which is not a manifold.

GKM actions on cohomogeneity one manifolds

We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group diagram of $M$ such that the $T$-action on $M$ is of GKM type, and describe its GKM graph. The general results are illustrated on explicit examples.

Cohomogeneity-One Quasi-Einstein Metrics

Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates on the rate of blow-up for these metrics near a singularity under a mild assumption on $G/H$. Next, we demonstrate that we can find quasi-Einstein metrics satisfying arbitrary $G$-invariant Dirichlet conditions.

Nonnegative curvature, low cohomogeneity and complex cohomology

We construct several infinite families of nonnegatively curved manifolds of low cohomogeneity and small dimension which can be distinguished by their cohomology rings. In particular, we exhibit an infinite family of eight-dimensional cohomogeneity one manifolds of nonnegative curvature with pairwise non-isomorphic complex cohomology rings.

Cohomogeneity one Einstein-Sasaki 5-manifolds

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.

具有两个求和项的余齐次1 Einstein度量的不存在性

Non-existence of cohomogeneity one Einstein metrics of two summands

针对主轨道含两个不等价不可约求和项的双圆盘丛紧流形，本文利用相空间势垒论证证明余齐次1 Einstein度量不存在，并提出新的存在性障碍。

The Dirichlet Problem for the Prescribed Ricci Curvature Equation on Cohomogeneity One Manifolds

Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci curvature equation $\mathrm{Ric}(G)=T$ for a Riemannian metric $G$ on $M$ subject to the boundary condition $G_{\partial M}=R$ (the notation $G_{\partial M}$ here stands for the metric induced by $G$ on $\partial M$). Imposing a standard assumption on $M_1$, we describe a set of requirements on $T$ and $R$ that guarantee global and local solvability.

Rational Ellipticity in Cohomogeneity Two

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rational numbers, is a finite-dimensional vector space over the rationals, then M is said to be rationally elliptic. It is known that M is rationally elliptic if it supports an action of cohomogeneity zero or one. When the cohomogeneity is two, this general result is no longer true. How…

Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).

紧致的cohomogeneity一 manifold上的不可压缩欧拉流体

Incompressible Euler fluids on compact cohomogeneity one manifolds

研究了紧致cohomogeneity一 manifold上的不可压缩欧拉流体，证明了任何G不变的无散向量场都能生成具有实数时间区间且光滑的流体速度-压力对。

Curvature homogeneous manifolds in dimension 4

We classify complete curvature homogeneous metrics on simply connected four dimensional manifolds which are invariant under a cohomogeneity one action. We show that they are either isometric to a symmetric space with one of its cohomogeneity one actions, or to a complete example by Tsukada on the normal bundle of the Veronese surface in CP^2. Along the way we show (in any dimension) that via an equivariant diffeomorphism the functions describing the metric can be partially diagonalized, a fact that may be useful for other problems as well

Special cohomogeneity one metrics with Q^111 or M^110 as principal orbit

We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.

Vector bundles of non-negative curvature over cohomogeneity one manifolds

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory…

G-manifolds with positive Ricci curvature and many isolated singular orbits

We show that in cohomogeneity 3 there are G-manifolds with any given number of isolated singular orbits and an invariant metric of positive Ricci curvature. We show that the corresponding result is also true in cohomogeneity 5 provided the number of singular orbits is even.