Examples of Riemannian Manifolds with non-negative sectional curvature

k-harmonic curves into a Riemannian manifold with constant sectional curvature

J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we describe the ordinary differential equations of $3$-harmonic curves into a Riemannian manifold with constant sectional curvature, and show biharmonic curve is k-harmonic curve $(k\geq 2)$.

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minim…

常截面曲率黎曼流形中的双调和Wintgen理想子流形

Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature

本文证明非正常截面曲率黎曼流形中的双调和Wintgen理想子流形是极小的，正常截面曲率情形下每个连通分量具有常平均曲率，部分肯定了Chen猜想、双曲空间中的广义Chen猜想以及球面中的Balmuş-Montaldo-Oniciuc猜想。

Sectional curvature for Riemannian manifolds with density

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth) 1/4-pinched sphere theorem. The main idea is to modify the radial curvature equation and second variation formula and then apply the techniques of classical Riemannian geometry to these new equations.

RC-positivity and rigidity of harmonic maps into Riemannian manifolds

In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.

Generalized Sasakian space forms and Riemannian manifolds of quasi constant sectional curvature

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $α$-Sasakian space form; or it is of five dimension and admits an $α$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A …

具有常截面曲率的流形上的λ-双和子浸入

λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature

本文研究了泛化双和子浸入的λ-双和子浸入，证明了从具有常截面曲率c的(n+1)维黎曼流形到n维流形的非存在性结果，关键值λ=2(n-1)c起决定性作用，当λ≠2(n-1)c时证明非存在性定理，而当λ=2(n-1)c时在非负曲率情况下证明非存在性定理，负曲率情况下构造了显式例子。

Moments of Sectional Curvature

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distribution of the sectional curvature of Riemannian products. Moreover we calculate the moments and the distribution of the sectional curvature for all compact symmetric spaces of rank 1 explicitly and derive a formula for the moments of general symmetric spaces. Interpolating the exp…

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We study the pseudohermitian sectional curvature of a CR manifold.

Almost Kähler manifolds of constant antiholomorphic sectional curvature

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

Mixed sectional-Ricci curvature obstructions on tori

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is…

厄米流形的陈截面曲率

The Chern Sectional Curvature of a Hermitian Manifold

本文定义了厄米流形的陈截面曲率，推导了其局部表达式，并证明当它与黎曼截面曲率一致时流形为凯勒流形，同时给出了里奇曲率和标量曲率。

Hermitian manifolds of pointwise constant antiholomorphic sectional curvatures

In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.

Symplectic mean curvature flows in Kähler surfaces with positive holomorphic sectional curvatures

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\geqδ$ is preserved along the flow.

A conclusive theorem on Finsler metrics of sectional flag curvature

If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.

On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

Einstein four-manifolds of pinched sectional curvature

In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming an upper bound on the difference between sectional curvatures.

On the boundedness of the sectional curvature of almost Hermitian manifolds

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold is of pointwise constant antiholomorphic sectional curvature. Similar results are obtained for almost Hermitian manifolds of definite metric.

k-harmonic maps into a Riemannian manifold with constant sectional curvature

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and show non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces, and study k-harmonic curve in Euclidean spaces. Futhermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.

Collapsed Riemannian manifolds with bounded sectional curvature

In the last two decades, one of the most important developments in Riemannian geometry is the collapsing theory of Cheeger-Fukaya-Gromov. A Riemannian manifold is called (sufficiently) collapsed if its dimension looks smaller than its actual dimension while its sectional curvature remains bounded (say a very thin flat torus looks like a circle in a bared eyes). We will survey the development of collapsing theory and its applications to Riemannian geometry since 1990. The common starting point for all of these is the existence of a singular fibration structure on collapsed manifolds. However, n…

A sectional curvature for statistical structures

A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.

具有非负截面曲率的四元Kähler流形

Quaternionic-Kähler manifolds with nonnegative sectional curvature

本文证明具有正数量曲率和非负截面曲率的紧四元Kähler流形等距于对称空间，推广了Berger的经典定理。

On almost quasi-negative holomorphic sectional curvature

A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact Kähler manifold admitting a Kähler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this paper we shall consider a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact Kähler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality $\int_Xc_1(K_X)^n>0$ in terms of the holomorphic sectional curvature. In the discussions, w…

Null Sectional Curvatures of Warped Products

In this paper, we investigate the null (light-like) sectional curvatures of Lorentzian warped product manifolds. We derive the formulas for the null sectional curvature of many well-known warped product space-time models such as multiply generalized Robertson-Walker space-times, generalized Kasner spacetimes and standard static space-times.

Nearly Kähler manifolds of constant antiholomorphic sectional curvature

A classification theorem for nearly Kähler manifolds of constant antiholomorphic sectional curvature is proved.

Sharp sectional curvature bounds and a new proof of The Spectral Theorem

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curvatures at a point. By extending our methods, we give a relatively short proof of the Spectral Theorem for self-adjoint operators on a finite dimensional real vector space.

Hypersurfaces with constant sectional curvature of $\Sf^n\times \R$ and $\Hy^n\times \R$

We classify the hypersurfaces of $\Sf^n\times \R$ and $\Hy^n\times \R$ with constant sectional curvature and dimension $n\geq 3$.

$4$-dimensional almost-Kähler Lie algebras of constant Hermitian holomorphic sectional curvature are Kähler

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

六维欧氏空间中的四维伪脐双调和子流形

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。

Remarks on symplectic sectional curvature

In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the symplectic connection and its covariant derivatives. Some relations between various more general conditions on the symplectic sectional curvature and the geometry of the symplectic connection or that induced on a symplectic submanifold are explored as well.

Positive weighted sectional curvature

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's fo…

Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks

This paper deals with the computation of sectional curvature for the manifolds of $N$ landmarks (or feature points) in D dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e. the cometric), when written in coordinates, is such that each of its elements depends on at most 2D of the ND coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly non-trivial problem of developing a formula that expresses sectional curvature in…

二维或三维洛伦兹流形的截面曲率与结构函数

Sectional Curvature and Structural Functions of a Two-Dimensional or Three-Dimensional Lorentzian Manifold

研究二维或三维光滑洛伦兹流形截面曲率，通过推导公式得出相关结果。

近切触度量流形上统计结构的ϕ-截面曲率

ϕ-Sectional curvature of statistical structures on almost contact metric manifolds

本文研究了近切触度量流形上统计结构诱导的ϕ-截面曲率，证明了其非正性，给出了曲率消失的等价条件，并推导出流形被归类为余辛统计流形的充分条件。

On Sectional Curvature Operator of 3-dimensional Locally Homogeneous Lorentzian Manifolds

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.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on Brieskorn quotients in dimension 5

We show that the moduli space of Riemannian metrics of nonnegative sectional curvature on certain quotients of Brieskorn varieties in dimension 5 has infinitely many path components. The same is true for the corresponding moduli space of positive Ricci curvature.

Some curvature pinching results for Riemannian manifolds with density

In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci solitons. In the case of sectional curvature we prove a new sphere theorem.

具有常全纯截面曲率的Kähler度量的局部分类

Local classification of Kähler metrics with constant holomorphic sectional curvature

该研究借助全纯函数1-节丛的几何特性，完成了常全纯截面曲率Kähler度量的局部分类证明，推进了复微分几何相关研究。

具有非负截面曲率的环面爱因斯坦4-流形

Toric Einstein 4-manifolds with nonnegative sectional curvature

本文证明了4-流形上具有非负截面曲率的$T^2$-不变爱因斯坦度量是局部对称的。

g-Natural metrics of constant sectional curvature

We prove that the tangent bundle endowed with a g-natural metrics has constant sectional curvature if and only if it is flat, and then we give a characterization of flat g-natural metrics on tangent bundles.

二维洛伦兹流形的截面曲率与结构函数

Sectional curvature and structure functions of a two-dimensional Lorentzian manifold

本文推导了任意二维洛伦兹流形上截面曲率的显式公式，为该类几何提供了统一计算工具。

Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of …

The weighted connection and sectional curvature for manifolds with density

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors …

AH3-manifolds of constant antiholomorphic sectional curvature

The following theorem is proved: If an AH3-manifold M of dimension greather or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then M is a real space form or a complex space form.

On Einstein Manifolds of Positive Sectional Curvature

Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.

The Holomorphic Sectional Curvature of General Natural KÄhler Structures on Cotangent Bundles

We study the conditions under which a Kählerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural Kählerian structure $(G,J)$.

Argument shift method and sectional operators: applications to differential geometry

This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively equivalent Riemannian and pseudo-Riemannian metrics.

Einstein Hermitian Metrics of Non Negative Sectional Curvature

We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.

Almost Kähler manifolds whose antiholomorphic sectional curvature is pointwise constant

It is proved that if an almost Kähler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.

Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatures.

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.

两个空间形式乘积中具有常截面曲率的超曲面

Hypersurfaces with constant sectional curvature of a product of two space forms

该研究完成了两个空间形式乘积中维数≥3的常截面曲率超曲面的分类

The vanishing rate of Weil-Petersson sectional curvatures

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with almost vanishing curvature. We bound the sectional curvature away from zero in terms of the product of lengths of short geodesics on Riemann surfaces. We give examples and an expectation for the actual vanishing rate.

Extreme values of sectional curvature on the homogeneous complex manifolds $U(n+1)/U(n)\times U(p+1)/U(p)$

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.