A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications
The author proves a splitting theorem for Alexandrov spaces of nonnegative curvature without properness assumptions [c.f., \textit{A. D. Milka}, Ukr. Geom. Sb. 4, 43--48 (1967; Zbl 0184.47404)]: Theorem. Let \(X\) be an Alexandrov space of nonnegative curvature. Then \(X\) is isometric to the direct product of some Hilbert space and some Alexandrov space with nonnegative curvature which contains no lines. As a corollary, a maximal radius theorem for Alexandrov spaces of curvature bounded from below is obtained without properness assumptions: Corollary. If an Alexandrov space \(X\) of curvature \(\geq 1\) has radius equal to \(\pi\), then \(X\) is isometric to the unit sphere of some Hilbert space with angle metric. Finally, new examples of infinite dimensional Alexandrov spaces of nonnegative curvature are found.
scientific article; zbMATH DE number 5675138