N. A. Daurtseva
Invariant complex structures on the homogeneous manifold $U(n+1)/U(n)\times U(p+1)/U(p)$ are reseached. Extreme values of sectional curvature of Hermitian metrics on this manifold are found.
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The main purpose of this paper is to determine the admissible forms of the sectional curvature operator on a three-dimensional locally homogeneous Lorentzian manifolds.
Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres
本文证明Ricci流在齐次球面和复射影空间上保持正截面曲率，结合先前结果完成了齐次空间中正曲率度量是否流出正曲率度量集的分类。
Fixed Point Homogeneous Orbifolds with Positive Sectional Curvature
本文对等变微分同胚下具有正截面曲率的不动点齐性黎曼轨形分类，进而得到具极大对称秩的正曲率轨形分类结果。
We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
The authors give a short survey of previous results on $δ$-homogeneous Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable $δ$-homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds $Sp(l)/U(1)\cdot Sp(l-1)=\mathbb{C}P^{2l-1}$…
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.
In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left invariant Randers metric of Berwald type has three dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.
It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or a product $\mathbb{R}\times N(c)$, where $N(c)$ is a space of constant curvature.
Contraction of hypersurfaces with positive sectional curvature in hyperbolic space
研究双曲空间中正截面曲率紧致超曲面的收缩曲率流，证明初始超曲面的正截面曲率在流中保持，并在有限时间内收缩至圆点。
We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one special Aloff Wallach space where the set of homogeneous metrics is too large).
The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on S^n bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include the first non-homogeneous examples of this type and certain positively curved Eschenburg and Aloff-Wallach spaces.
It is well known that simply connected symmetric spaces of non-positive sectional curvature admit a linear isoperimetric filling inequality for cycles of dimension greater than or equal to the rank of the space. In this note we extend that result to homogeneous Hadamard manifolds.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions, we do not know if they admit homogeneous Finsler metrics with positive Flag curvature.
We show that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.
We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of $CP^n$, $HP^n$ and $CaP^2$, and a family of lens space bundles over $CP^n$. All new examples are consequences of a general sufficient condition for a homogeneous fiber bundle over a homogeneous space to admit such a metric.
In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrast with the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant secti…
We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respectively. In addition the corresponding homogeneous models are computed, exhibiting the relation between these kind of structures and the incompleteness of the metric.
Hypersurfaces of manifolds of constant nonzero sectional curvature are classificated according their restricted homogeneous holonomy groups.
We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product $\mathbb S^1 \times M$ carries canonical families of Weyl connections with such a property, for any Riemmanian manifold $M$. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (streched non-positive), then it must be locally of the product type.
For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature.
We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, and that most compact homogeneous spaces with positive sectional curvature satisfy it.
Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal sphere to some component of $Fix(M,G)$, equivalently, if $Fix(M,G)$ has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.
Nonexistence of stable discrete maps into some homogeneous spaces of nonnegative curvature
本文研究了从有限加权图到光滑黎曼流形的离散映射的加权长度或能量泛函的稳定性，并证明了到特定齐性空间（如正全纯截面曲率的Kähler C-空间及部分单连通紧致黎曼对称空间）不存在稳定离散极小浸入或非常数稳定离散调和映射。
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetri…
Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.
Lorentzian homogeneous structures with indecomposable holonomy
本文研究洛伦兹齐性空间中迷向群的代数条件对几何与曲率的影响，证明具有不可分解、非不可约和乐群的Ambrose--Singer联络的洛伦兹局部齐性空间局部等距于平面波，推广了已有结果并给出关于平行挠率联络与2-对称联络的新结论。
The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the isometry group of some of the biquotients due to Eschenburg and Bazaikin. As an application we obtain, as in the homogeneous case, more counterexamples to the Chern conjecture, which states that an abelian subgroup of the fundamental group of a positively curved manifold is cyclic.
In this paper, we consider half-flat $SU(3)$-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form $w_1^-$ is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non coupled and non double) structure inducing an Einstein metric on $S^3\times S^3$ and we show there does not exist any left invariant coupled structure inducing an ${\rm Ad}(S^1)$-invariant Einstein metric on it. Fina…
Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$
本文通过 Hopf 纤维定义 homogeneous nearly Kähler 结构，并研究了 $\mathbb{C}P^3$ 中 Lagrangian 子流形的曲率性质与等距群。
Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.
If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that decay like the inverse of t and diameter that increases at most like the square root of t, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.
Weakly holomorphic homogeneous regular manifolds
本研究提出弱全纯齐次正则流形（wHHR）这一新复流形类别，证明其包含全纯齐次正则流形等多类流形，并验证wHHR Stein流形上Bergman与Kobayashi度量双Lipschitz等价。
We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the wo…
Positive $k^{\rm th}$-intermediate Ricci curvature on a Riemannian $n$-manifold, to be denoted by $\mathrm{Ric}_k > 0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when $k =1$ and $k=n-1$ respectively). In this work, we produce many examples of manifolds of $\mathrm{Ric}_k > 0$ with $k$ small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $n\geq 7$ congruent to $3\,\mathrm{mod}\ 4$ supports infinitely many closed simply connected…
The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant completely non- holonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin-Hamilton func- tion. This function was ap…
Cohomogeneity two Bazaikin spaces
该研究针对Wilking的黎曼度量构造，分析上同调数为2的Bazaikin空间的截面曲率，发现这类空间中严格正曲率的点集不具有满测度，与上同调数为1及齐次的情况形成对比。
Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characterized up to local isometry by their spectra. We observe that there are eight `metrically maximal' three-dimensional geometries on which all compact locally homogeneous three-manifolds are modeled and we demonstrate that for five of these geometries the associated compact locally …
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…
Examples of almost-positively and quasi-positively curved spaces of the form M=H((G,h)xF) were discovered recently. Here, h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H of G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G,h)xF with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F={point} proven by Geroch.
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.
In this note we study globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ of homogeneous Riemannian manifolds $(M,ds^2)$. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$. We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition ca…
We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut $δ$-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any $δ$-homogeneous Riemannian manifolds is itself $δ$-homogeneous. In turn, every simply connected Riemannian $δ$-homogeneous manifold is a direct metric product of an Euclidean space and compact simply connected indecomposable homogeneous manifolds; all factors in this product are itself $δ$-homogeneous. We find different characterizations of $δ$-homogeneous Riemannian spac…
We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally…
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。
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.
The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces
本文研究了紧致自然红uctive齐性空间上拉普拉斯-贝叶斯算子的谱，通过广义Casimir算子的特征值和球形表示推导出公式，并应用于正常齐性度量的变体，首次分析了谱对度量变形参数的依赖性。
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…
Non-negative curvature on certain product manifolds
本文证明：若闭单连通齐性流形 G/H 的实向量丛稳定类均含齐性丛，则任何与其同伦等价的闭单连通光滑流形 M 与某高维球面的乘积容许非负截面曲率度量。
We introduce a new notion of a homogeneous pair for a pseudo-Riemannian metric $g$ and a positive function $f$ on a manifold $M$ admitting a free $\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For example, (a) a class of pseudo-Hessian manifolds admitting a free $\mathbb{R}_{>0}$-action and a homogeneous potential function such as the moduli space of torsion-free $G_2$-structures, (b) the space of Riemannian metrics on a compact manifold, and (c) many moduli spaces of geometric structures such as torsion-free ${\rm Spin}(7)$-structures admit this structure. Hence w…
Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotiente $\widetilde{N}^{11}=(Sp(2)\times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotiente $\widetilde{N}^{11}/S^3$, we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere $Σ^7$. Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature…
Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching
研究\(n\)维欧几里得向量空间上代数曲率张量的尖锐估计，通过应用该估计到流形曲率张量分解，在特定挤压条件下得出\(H^2(M;\mathbb{R})\)消失等结论，还给出截面 - 标量挤压准则及相关流形性质。
In this note we complete a study of globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ in positive curvature. Specifically, $M$ is a homogeneous space $G/H$ that admits a $G$-invariant Riemannian metric of strictly positive sectional curvature, and $ds^2$ is a $G$--invariant Riemannian metric on $M$, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$. In an earlier paper we verified tha…
The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\R^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact ran…
Non-homogeneous curvature flows in hyperbolic space
本文研究双曲空间中非齐次曲率流，证明在权函数结构条件下流光滑存在、保持严格h-凸性并收缩至固定点，且归一化径向函数指数收敛。
This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, we prove that for the spaces $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$ and $F_4/Spin(8)$, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricc…
The Lie group SO_0(n, 1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left. The geometry of the quotient space of the homogeneous submersion SO_0(n, 1) -> SO(n)\SO_0(n, 1) is investigated. The space is expressed as a warped product. Its group of isometries and sectional curvatures are calculated.
Curvatures of Stiefel manifolds with deformation metrics
本文计算了Stiefel流形上一族包含嵌入度量和典范度量的Cheeger变形度量的曲率，确定了Einstein度量的参数条件，分析了截面曲率范围并给出列数为2时的精确结果。
The Bishop-Gromov theorem is a comparison theorem of differential geometry that upperbounds the growth of volume of a geodesic ball in a curved space. For many spaces, this bound is far from tight. We identify a major reason the bound fails to be tight: it neglects the effect of shear. By using higher curvature invariants to lowerbound the average shear, we are able to place tighter-than-Bishop-Gromov upperbounds on the late-time growth rates of geodesic balls in homogeneous spaces with non-positive sectional curvature. We also provide concrete guidance on how our theorem can be generalized to…
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