self dual对称R-空间中的圆

摘要：

自对偶对称R-空间有特殊的曲线，称为圆，由Burstall、Donaldson、Pedit和Pinkall在2011年引入，其定义不涉及任何黎曼度量的选择。本文将自对偶对称R-空间M的大变换群G的元素刻划为M在圆中发送圆的那些同态。此外，虽然这些曲线属于G的不变量的范围，但我们设法用黎曼几何术语来描述它们：给定M中的一个圆c，存在G的一个极大紧子群K，使得c，除了射影重参数化之外，是M中的一条直径测地线（或等价地，M的极大全测地平坦环面中的对角测地线），只要M携带标准对称K-不变度量。我们包括复杂的二次曲面和分裂标准或各向同性格拉斯曼的例子。

英文摘要：

Self dual symmetric R-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group G of a self dual symmetric R-space M as those diffeomorphisms of M sending circles in circles. Besides, although these curves belong to the realm of the invariants by G, we manage to describe them in Riemannian geometric terms: Given a circle c in M, there is a maximal compact subgroup K of G such that c, except for a projective reparametrization, is a diametrical geodesic in M (or equivalently, a diagonal geodesic in a maximal totally geodesic flat torus of M), provided that M carries the canonical symmetric K-invariant metric. We include examples for the complex quadric and the split standard or isotropic Grassmannians.

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