PreprintCC BY 4.0arXiv
Cyclic Riemannian nilmanifolds are naturally reductive
Abstract
We study the structure of cyclic Riemannian nilmanifolds, that is, connected nilpotent Lie groups N endowed with a left-invariant cyclic metric ⟨·,·⟩. We prove that a Riemannian nilmanifold (N,⟨·,·⟩) is cyclic if and only if its Lie algebra n is at most two-step nilpotent and the associated family of skew-symmetric endomorphisms Jz={JZ : Z∈ z}⊂ so(a) is Abelian. As a direct consequence, every cyclic Riemannian nilmanifold is naturally reductive. We also determine the full isometry group of a connected and simply connected cyclic Riemannian nilmanifold without Euclidean factor.
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BibTeX
@misc{wang2026cyclic,
title = {{Cyclic Riemannian nilmanifolds are naturally reductive}},
author = {Hui Wang and Zaili Yan and Shaoxiang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16457},
eprint = {2609.16457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cyclic Riemannian nilmanifolds are naturally reductive
AU - Hui Wang
AU - Zaili Yan
AU - Shaoxiang Zhang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16457
UR - https://arxiv.org/abs/2609.16457
AB - We study the structure of cyclic Riemannian nilmanifolds, that is, connected nilpotent Lie groups N endowed with a left-invariant cyclic metric ⟨·,·⟩. We prove that a Riemannian nilmanifold (N,⟨·,·⟩) is cyclic if and only if its Lie algebra n is at most two-step nilpotent and the associated family of skew-symmetric endomorphisms J_z={J_Z : Z∈ z}⊂ so(a) is Abelian. As a direct consequence, every cyclic Riemannian nilmanifold is naturally reductive. We also determine the full isometry group of a connected and simply connected cyclic Riemannian nilmanifold without Euclidean factor.
ER -