Closed counterexamples to Toponogov's question on mixed curvature

Abstract

We construct explicit closed Riemannian manifolds in every even dimension carrying one-dimensional totally geodesic foliations with positive mixed sectional curvature. The leaves are closed geodesics and form smooth circle fibrations. In dimensions $4m+2$ the examples are constructed on $S^{4m-1}\times S^3$; in dimensions $4m$ they are obtained as totally geodesic fixed point submanifolds $S^{4m-1}\times S^1$. Since the normal rank is odd in all examples, the Ferus--Adams bound predicted by Toponogov's question would force the leaf dimension to be zero. Thus positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds. The metrics are not bundle-like. Moreover, the examples can be chosen with mixed sectional curvature arbitrarily close to $1$-pinched, and after normalizing $\max K_{\mathrm{mix}}=1$, the lengths of the leaves remain uniformly bounded. The mechanism is that the positive mixed curvature operator rotates in a parallel normal frame, preventing the scalar Riccati reduction used in the constant-curvature case.