Geodesic orbit pseudo-Riemannian H-type nilmanifolds
Abstract
A pseudo-Riemannian manifold is called geodesic orbit (GO) if every geodesic is an orbit of a one-parameter group of isometries. For Riemannian H-type groups, the GO property was completely characterized by A. Kaplan and C. Riehm in terms of the dimension of the centre and the structure of the underlying Clifford module. We solve the corresponding problem for pseudo H-type groups Nr,s, that is, 2-step nilpotent Lie groups whose Lie algebra nr,s=z⊕v is built from an admissible module v of the Clifford algebra Cl(ℝr,s), endowed with the left-invariant metric whose restriction to the centre is indefinite (s≥ 1). We prove that Nr,s is naturally reductive if and only if (r,s)∈{(0,1),(1,2)}, and then it is GO for every admissible module. We also prove that N3,4 is GO, but not naturally reductive, exactly when v is a minimal admissible module, and that in all remaining cases Nr,s is not GO. Thus, in contrast with the Riemannian situation, where a seven-dimensional centre admits GO metrics on isotypic modules of dimensions 8, 16 and 24, the exceptional pseudo-Riemannian case is GO only for the smallest possible module. For N3,4(vmin) we explicitly construct the one-parameter isometry groups generating all geodesics, including the null ones, and describe the affine family of their generators.
Related
- Pseudo-Riemannian geodesic orbit nilmanifolds of signature (n−3,3)
- On the Structure of Sub-Riemannian Geodesic Orbit Manifolds
- Cyclic Riemannian nilmanifolds are naturally reductive
- A smooth Riemannian metric on an arbitrary product of spheres with Bernoulli geodesic flow
- Riemannian convex bodies with a geodesic segment on the boundary
BibTeX
@misc{furutani2026geodesic,
title = {{Geodesic orbit pseudo-Riemannian $H$-type nilmanifolds}},
author = {Kenro Furutani and Irina Markina and Yurii Nikonorov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06375},
eprint = {2610.06375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06375},
}
RIS
TY - UNPB
TI - Geodesic orbit pseudo-Riemannian H-type nilmanifolds
AU - Kenro Furutani
AU - Irina Markina
AU - Yurii Nikonorov
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06375
UR - https://arxiv.org/abs/2610.06375
AB - A pseudo-Riemannian manifold is called geodesic orbit (GO) if every geodesic is an orbit of a one-parameter group of isometries. For Riemannian H-type groups, the GO property was completely characterized by A. Kaplan and C. Riehm in terms of the dimension of the centre and the structure of the underlying Clifford module. We solve the corresponding problem for pseudo H-type groups N_r,s, that is, 2-step nilpotent Lie groups whose Lie algebra n_r,s=z⊕v is built from an admissible module v of the Clifford algebra Cl(ℝ^r,s), endowed with the left-invariant metric whose restriction to the centre is indefinite (s≥ 1). We prove that N_r,s is naturally reductive if and only if (r,s)∈{(0,1),(1,2)}, and then it is GO for every admissible module. We also prove that N_3,4 is GO, but not naturally reductive, exactly when v is a minimal admissible module, and that in all remaining cases N_r,s is not GO. Thus, in contrast with the Riemannian situation, where a seven-dimensional centre admits GO metrics on isotypic modules of dimensions 8, 16 and 24, the exceptional pseudo-Riemannian case is GO only for the smallest possible module. For N_3,4(v_min) we explicitly construct the one-parameter isometry groups generating all geodesics, including the null ones, and describe the affine family of their generators.
ER -