Closed counterexamples to Toponogov's question on mixed curvature

Abstract

We give a negative answer to Toponogov's question whether positive mixed sectional curvature on a closed manifold forces the Ferus-Adams dimension bound for a totally geodesic foliation. For every even $N\ge 4$, we construct explicit metrics on $S^{4m-1}\times S^3$ and on totally geodesic fixed point submanifolds $S^{4m-1}\times S^1$. The leaves are closed geodesics forming smooth circle fibrations, but the metrics are not bundle-like. One subfamily has a positive mixed-curvature operator that is parallel along every leaf but nonscalar, with spectrum $\{\kappa ,4\kappa \}$; it is a closed realization of Rovenskii's local anisotropic Riccati model. In a second subfamily, ${\delta }_{\mathrm{mix}}\to 1$ while the curvature operator remains nonparallel. After normalizing $\max K_{\mathrm{mix}}=1$, the leaf lengths tend to $2\pi$, thereby answering the bounded-length question for the local pinched models in the closed setting. Thus neither leafwise parallelism nor pinching by any fixed constant below one replaces scalarity in the Ferus argument; by contrast, the Ferus-Adams bound remains valid in the bundle-like case.

21 pages. Substantially revised and shortened for conceptual clarity. This replacement is a conceptual and expository revision; the main results of v1 are unchanged. Added a parallel-anisotropic subfamily, an explicit proof of the bundle-like case, and a sharper comparison with earlier local models