Sectional Curvature Pinching of Two-Step Nilmanifolds
Abstract
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.
Related
- Constant Chern holomorphic sectional curvature on two-step solvmanifolds
- Curvature pinching for minimal Tangent-Yang-Mills hypersurfaces in spheres
- Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow
- Sharp scalar and Ricci curvature pinching theorems for Lagrangian submanifolds in the homogeneous nearly Kähler Six-sphere
- Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
BibTeX
@misc{tatsuno2026sectional,
title = {{Sectional Curvature Pinching of Two-Step Nilmanifolds}},
author = {Tomoya Tatsuno},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13052},
eprint = {2609.13052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13052},
}
RIS
TY - UNPB
TI - Sectional Curvature Pinching of Two-Step Nilmanifolds
AU - Tomoya Tatsuno
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13052
UR - https://arxiv.org/abs/2609.13052
AB - 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 Heis_3(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 Heis_3(ℝ)× ℝ^n−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 Heis_3(ℝ)× ℝ^n−3, then there is a curve g_t of metrics on N with (N,g_t)∈ 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.
ER -