希尔伯特H型群中的Michor-Mumford猜想
The Michor-Mumford conjecture in Hilbertian H-type groups
AI总结：
本文引入无限维希尔伯特H型群并研究其弱左不变黎曼度量，证明测地距离消失与截面曲率局部无界性共存，验证Michor-Mumford猜想，还揭示退化测地距离与无限维子黎曼结构的关联及Levi-Civita导数的不存在性。
摘要：
我们引入了配备弱、分次、左不变黎曼度量的无限维希尔伯特H型群。对于这些李群，我们证明了测地距离消失与截面曲率局部无界性共存。该结果验证了Michor和Mumford在2005年一篇有影响力的论文中猜想的深层现象，即测地距离消失与截面曲率局部无界性相关。我们证明退化测地距离在一大类弱左不变黎曼度量中出现，其消失意外地与希尔伯特H型群的无限维子黎曼结构相关。同一类弱黎曼度量导致Levi-Civita协变导数不存在。
英文摘要：
We introduce infinite dimensional Hilbertian H-type groups equipped with weak, graded, left invariant Riemannian metrics. For these Lie groups, we show that the vanishing of the geodesic distance and the local unboundedness of the sectional curvature coexist. The result validates a deep phenomenon conjectured in an influential 2005 paper by Michor and Mumford, namely, the vanishing of the geodesic distance is linked to the local unboundedness of the sectional curvature. We prove that degenerate geodesic distances appear for a large class of weak, left invariant Riemannian metrics. Their vanishing is rather surprisingly related to the infinite dimensional sub-Riemannian structure of Hilbertian H-type groups. The same class of weak Riemannian metrics yields the nonexistence of the Levi-Civita covariant derivative.
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