Explicit constructions of closed even-dimensional manifolds with positive mixed sectional curvature on 1D totally geodesic circle foliations provide counterexamples to Toponogov's question by violating the predicted Ferus-Adams leaf dimension bound.
The fundamental equations of a submersion
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The Extended Real Line with Reentry is the first compact path-connected US-not-KC space, obtained via a density-modified quotient of the extended reals.
H-structures on closed manifolds are homotopy equivalent to their isometric classes via surjective metric map with lifting property, reducing to mapping spaces on parallelizable manifolds like tori, with applications to torsion energy flows.
Hydrodynamic limit of symmetric exclusion process on Poisson graphs approximating weighted Riemannian manifolds and principal bundles is a Fokker-Planck equation.
All computable increasing sequences converging to random reals converge at the same speed up to a constant factor, as this follows immediately from Bishop's upcrossing inequality.
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