We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these 2-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates how “rubber-band” topology can, in the presence of symmetry, single out very rigid objects. We characterize when these 2-spheres in the same homotopy class have the same geometry for all invariant metrics. This is done by exploring the action of Weyl group of the flag manifold, generalizing results of Patrão and San Martin (Indag. Math. 26:547–579, 2015) and de Siebenthal (Math. Helvetici 44(1):1–3, 1969). This illustrates how some aspects of “continuum” invariant geometry can, in the presence of symmetry, be reduced to the study of discrete objects. We remark that the topology singling out very rigid objects and the study of a continuum object being reduced to discrete ones is a characteristic of situations with a lot of symmetry and, thus, are recurring themes in Lie theory.

中文翻译：

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标志流形的第二同伦群与不变几何

我们使用 Hopf fibration 来显式计算紧李群的标志流形的第二同伦群的生成器。我们展示了这些 2 球体具有良好的几何特性，例如对于标志流形上的任何不变度量，都是完全测地曲面，概括了 Burstall 和 Rawnsley 的结果（Springer Lect. Notes Math. 2(84):1424, 1990 ）。这说明了“橡皮筋”拓扑如何在存在对称性的情况下挑选出非常刚性的物体。我们表征当同一同伦类中的这些 2 球体对于所有不变度量具有相同的几何形状时。这是通过探索标志流形的 Weyl 群的作用，概括 Patrão 和 San Martin (Indag. Math. 26:547–579, 2015) 和 de Siebenthal (Math. Helvetici 44(1):1–3) 的结果来完成的, 1969)。这说明了“连续”不变几何的某些方面如何在存在对称性的情况下被简化为对离散对象的研究。我们注意到拓扑分离出非常刚性的物体以及将连续体物体简化为离散物体的研究是具有大量对称性的情况的特征，因此是李理论中反复出现的主题。