Abstract We show, in contrast to the cases of complex and quaternionic partial flag manifolds, the rational cohomology rings of the real and oriented partial flag manifolds are generated not only by the characteristic classes of their canonical vector bundles, but also by exterior algebras arising from the rational cohomology rings of real Stiefel manifolds. We also discuss the rational and the mod-2 equivariant cohomology rings of these partial flag manifolds for certain canonical actions of tori and 2-tori.

中文翻译：

---

实数和定向部分标志流形的上同调环

摘要 我们表明，与复数部分标志流形和四元数部分标志流形的情况相比，实部和有向部分标志流形的有理上同调环不仅由其典型向量丛的特征类产生，而且由来自实 Stiefel 流形的有理上同调环。我们还讨论了这些部分标志流形的有理环和 mod-2 等变上同调环，用于 tori 和 2-tori 的某些规范动作。