Nonnegative curvature, symmetry and fundamental group (Q1771156)

\textit{B. Wilking} proved that given a compact Riemannian manifold \((M,g)\) of Ricc; curvature \(\text{Ric} \geq 0\) there exists a continuous family of metrics \(g_t, t\in [0,1]\) on \(M\) with \(g_0=g\) such that the universal covers of \((M,g_t)\) are all isometric and a finite cover of \((M,g_1)\) is isometric to the Riemannian product of a flat torus and a simply connected \(N,\) see [Differ. Geom. Appl. 13, No. 2, 129--165 (2000; Zbl 0993.53018)]. This sharpened results of Cheeger and Gromoll. The author shows that for infinite \(\pi_1(M)\) and a compact subgroup \(H\) of \(\text{Iso} (M,g)\) the deformation \(g_t\) can be chosen to satisfy the condition \(H\leq \text{Iso} (M,g_t), t\in [0,1].\) A similar theorem is proved for complete \(M\) of \(\text{sec} \geq 0.\) These statements are derived from a result on the equivariant deformations of flat bundles. As a consequence the author constructs examples of Lie group actions on \(M\) with infinite \(\pi_1(M)\) that are isometric in some metric of positive scalar curvature and not isometric in any metric of \(\text{Ric} \geq 0\). Similar examples are constructed with the condition \(\text{Ric} \geq 0\) replaced by \(\text{sec}\geq 0\) and without the assumption \(\pi_1(M)\) being infinite.