Non-singular weakly symmetric nilmanifolds
Y. Nikolayevsky, W. Ziller
TL;DR
This work delivers a complete classification of simply connected non-singular weakly symmetric nilmanifolds arising from 2-step nilpotent Lie groups by organizing them into WS-pairs (V, ⟨·,·⟩) with V ⊂ so(𝔞). Building on GO-pair classifications, the authors show every non-singular WS-pair is either of Clifford type or lies in a finite set of explicit non-Clifford families, including a 1-parameter 14-dimensional family and a 3-dimensional center case, with deformations of Clifford-type capturing the WS examples. The core method blends nilpotent Lie algebraic structure with symmetry analysis via the normalizer 𝔑(V) and centralizer 𝔠(V), establishing when the WS condition can be satisfied and ruling out WS in non-WS GO-pairs. The results advance the understanding of weak symmetry in nilmanifolds, connect to commutative nilmanifold classifications, and map out the landscape of possible WS geometries in the 2-step nilpotent setting, including several new families and parameterized deformations.
Abstract
A Riemannian manifold $M$ is called weakly symmetric if any two points in $M$ can be interchanged by an isometry. The compact ones have been well understood, and the main remaining case is that of 2-step nilpotent Lie groups. We give a complete classification of simply connected non-singular weakly symmetric nilmanifolds. Besides previously known examples, there are new families with 3-dimensional center, and a one-parameter family of dimensions 14. The classification is based on the authors classification of non-singular 2-step nilpotent Lie groups for which every geodesic is the image of a one parameter group of isometries.