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Search: All articles in the CMB digital archive with keyword flag

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1. CMB Online first

Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya
The equivariant cohomology rings of Peterson varieties in all Lie types
Let $G$ be a complex semisimple linear algebraic group and let $Pet$ be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring $H^*_S(Pet)$ of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type $A$.

Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula
Categories:55N91, 14N15

2. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature
Categories:17B20, 22E46, 53C12

3. CMB 2011 (vol 55 pp. 474)

Chen, Bin; Zhao, Lili
A Note on Randers Metrics of Scalar Flag Curvature
Some families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic $S$-curvature are given. Certain Randers metrics with Einstein $\alpha$ are considered and proved to be complex. Three dimensional Randers manifolds, with $\alpha$ having constant scalar curvature, are studied.

Keywords:Randers metrics, scalar flag curvature
Categories:53B40, 53C60

4. CMB 2009 (vol 53 pp. 122)

Mo, Xiaohuan; Zhou, Linfeng
A Class of Finsler Metrics with Bounded Cartan Torsion
In this paper, we find a class of $(\alpha,\beta)$ metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.

Keywords:Finsler manifold, $(\alpha,\beta)$ metric, Cartan torsion, R-quadratic, flag curvature
Category:58E20

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