论文标题
关于脂肪捆的垂直曲率的注释
A note on the vertizontal curvature of fat bundles
论文作者
论文摘要
In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical curvatures equal to $ 1 $？在本说明中，我们为具有有限的载体和特定曲率约束的脂肪里曼植物建立了刚性结果。我们的结果解决了齐勒（Ziller）对脂肪纤维束的问题，该问题考虑了在曲率约束下连接的紧凑总空间，该曲率约束（例如局部对称空间）。此外，我们假设所有垂直曲率都在某个点重合。
In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical curvatures equal to $1$? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.
