## Abstract

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define the property “naturally reductive” for pseudo-Riemannian manifolds and note that those manifolds are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular that result applies to pseudo-Riemannian Lorentz manifolds. Our main results are Theorems 7 and 8. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or 2-step nilpotent. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds *G*/*H* with \(G = N \rtimes H\) and *N* nilpotent. When the metric is nondegenerate on \([{{\mathfrak {n}}},{{\mathfrak {n}}}]\), Theorem 7 shows that *N* either is at most 2-step nilpotent as in the Riemannian situation, or is 4-step nilpotent, but cannot be 3-step nilpotent. Surprisingly, Theorem 8 shows that *N* is at most 2-step nilpotent when the metric is degenerate on \([{{\mathfrak {n}}},{{\mathfrak {n}}}]\). Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds.

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## Notes

Selberg: Let (

*M*,*g*) be a Riemannian manifold. If there exists a subgroup*G*of the isometry group*I*(*M*,*g*) of*M*acting transitively on*M*and an involutive isometry \(\mu \) of (*M*,*g*) with \(\mu G = G \mu \) such that whenever \(x, y \in M\) there exists \(\phi \in G\) with \(\phi (x) = \mu (y)\) and \(\phi (y) = \mu (x)\), then (*M*,*g*) is a*weakly symmetric Riemannian manifold*.

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## Acknowledgements

ZC was partially supported by National Natural Science Foundation of China (11931009 and 12131012) and Natural Science Foundation of Tianjin (19JCYBJC30600). JAW was partially supported by the Simons Foundation. SZ was partially supported by National Natural Science Foundation of China (12071228 and 51535008), Natural Science Foundation of Shandong Province (ZR2021QA051), and SZ thanks the China Scholarship Council for support at University of California, Berkeley and he also thanks U. C. Berkeley for hospitality.

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Chen, Z., Wolf, J.A. & Zhang, S. On the Geodesic Orbit Property for Lorentz Manifolds.
*J Geom Anal* **32**, 81 (2022). https://doi.org/10.1007/s12220-021-00744-8

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DOI: https://doi.org/10.1007/s12220-021-00744-8

### Keywords

- Geodesic orbit space
- Lorentz nilmanifold
- Weakly symmetric space
- Naturally reductive space
- Pseudo-Riemannian manifold