There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.

中文翻译：

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山雀几何形状和正曲率

Dadok 提供的对称空间的（最大）极坐标表示和各向同性表示之间存在众所周知的联系。此外，Tits 和 Burns-Spatzier 的理论提供了阶数至少为 3 的非紧凑型不可约对称空间与阶数至少为 3 的不可约拓扑球形建筑物之间的联系。我们发现并利用了有限 (Coxeter) M 型（连接）室系统的丰富结构，该系统与任何简单连接的闭合正弯曲歧管上至少两个同质性的任何极性作用相关联。虽然这个房间系统通常不是 M 类型的 Tits 几何结构，但我们证明在所有情况下，除了两种情况外，其通用 Tits 覆盖物确实是一座建筑物。我们在这个通用封面上构建了一个拓扑结构，使其成为 Burns 和 Spatzier 意义上的紧凑球形建筑。