https://neckpinch.com/authors/tomoya-tatsuno/New arXiv papers by Tomoya Tatsuno, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2609.13052/arxiv:2609.13052Fri, 11 Sep 2026 00:00:00 GMTTomoya Tatsuno — We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval [−3, −3/2]. The upper bound −3/2 is achieved by the complex Heisenberg group Heis₃(ℂ) with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold N has the pinching constant −3/2, then N admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant −3. This is derived by showing that there is an open neighborhood U of Heis₃(ℝ)× ℝⁿ⁻³ in the space of n-dimensional 2-step nilmanifolds such that the pinching constant is −3 on U, and any 2-step nilpotent Lie group N has a metric g such that (N,g) lies in U. In fact, if N is not isomorphic to Heis₃(ℝ)× ℝⁿ⁻³, then there is a curve gₜ of metrics on N with (N,gₜ)∈ U, showing that there are uncountably many left-invariant metrics on N such that the pinching constant is −3. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant −3/2 is also given, and the pinching constants of various examples are computed.Ricci solitonsCurvature pinchingHomogeneous spacesStability & uniqueness