Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.     

Equigeodesics on Generalized Flag Manifolds with b 2 (G / K) = 1

In this paper we provide a characterization of structural equigeodesics on generalized flag manifolds with second Betti number b ₂(G / K) = 1, and give examples of structural equigeodesics on generalized flag manifolds of the exceptional Lie groups F ₄, E ₆ and E ₇ with three isotropy summands.     

Odd symplectic flag manifolds

We define the odd symplectic Grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group, analogous to the usual symplectic Grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic Grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic Grassmannians and we prove a Borel-Weil-type theorem for the odd symplectic group.     

Invariant Hermitian structures and variational aspects of a family of holomorphic curves on flag manifolds

In this article, we study variational aspects for a special class of holomorphic maps on flag manifolds. We prove stability and non-stability results for such maps with respect to a large number of invariant Hermitian structures on maximal flag manifolds including the (1, 2)-symplectic structures, Einstein metrics, Cartan–Killing metrics, and so on. Some results in this article were announced without proofs in a past article by the first author.     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.     

Classical Symmetries of Complex Manifolds

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.     

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the Fomin–Kirillov algebra implies the nonnegativity of the Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.     

On a class of symmetric CR manifolds

We study and classify a large class of minimal orbits in complex flag manifolds for the holomorphic action of a real Lie group. These orbits are all symmetric CR spaces for the restriction of a suitable class of Hermitian invariant metrics on the ambient flag manifold. As a particular case we obtain that the standard compact homogeneous CR manifolds associated with semisimple Levi-Tanaka algebras are symmetric CR-spaces.     

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.     

李群表示论和Schubert条件

本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答.     

满旗流形SO(8)=T上不变爱因斯坦度量

本文研究了迷向表示分为12个不可约子空间的满旗流形SO(8)=T上不变爱因斯坦度量的问题.利用计算机计算满旗流形SO(8)=T爱因斯坦方程组的方法, 得到了满旗流形SO(8)=T上有160 个不变爱因斯坦度量(up to a scale)的结果, 在等距情况下考虑这160个不变爱因斯坦度量, 其中1个是凯莱爱因斯坦度量, 4 个是非凯莱爱因斯坦度量. 推广了只对迷向表示分为小于等于6个不可约子空间的满旗流形上不变爱因斯坦度量的研究.     

具有相对迷向平均Landsberg曲率的流形的一个整体刚性定理

讨论了具有相对迷向平均Landsberg曲率的度量的一些几何性质.证明了任一闭的具有负旗曲率与相对迷向平均Landsberg曲率的流形一定是Riemann流形.     

The Gelfand-Cetlin system and quantization of the complex flag manifolds

The construction of angle action variables for collective completely integrable systems is described and the associated Bohr-Sommerfeld sets are determined. The quantization method of Sniatycki applied to such systems gives formulas for multiplicities. For the Gelfand-Cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations.     

The topology of indefinite flag manifolds

Summary In this paper, we will use some techniques in Morse Theory in order to compute the Betti numbers of an indefinite flag manifold. The problem is reduced to compute it for the definite flag manifolds.     

Manifolds with 1/4-Pinched Flag Curvature

We say that a nonnegatively curved manifold (M, g) has quarter-pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter-pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd-dimensional manifolds we obtain results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent work of Petersen and Tao.     

The Smoothness of Laws of Random Flags and Oseledets Spaces of Linear Stochastic Differential Equations

The Oseledets spaces of a random dynamical system generated by a linear stochastic differential equation are obtained as intersections of the corresponding nested invariant spaces of a forward and a backward flag, described as the stationary states of flows on corresponding flag manifolds. We study smoothness of their laws and conditional laws by applying Malliavin's calculus. If the Lie algebras induced by the actions of the matrices generating the system on the manifolds span the tangent spaces at any point, laws and conditional laws are seen to be C-smooth. As an application we find that the semimartingale property is well preserved if the Wiener filtration is enlarged by the information present in the flag or Oseledets spaces.     

A global rigidity theorem for weakly Landsberg manifolds

In this paper we study a global rigidity property for weakly Landsberg manifolds and prove that a closed weakly Landsberg manifold with the negative flag curvature must be Riemannian.     

Twisted signatures of flag manifolds

A product formula for some twisted signatures of flag manifolds is proved. The result is used to compute twisted signatures of some flag manifolds from those of Grassmannians, and by that to deduce some upper bounds of the stable span.