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Examples of pseudo-Riemannian g.o. manifolds. (English) Zbl 1122.53030
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 144-155 (2007).

A Riemannian homogeneous manifold $\left(M=G/H,g\right)$ is called a g.o. manifold if every geodesic of $M$ is homogeneous, that is, an orbit $\gamma \left(t\right)=exptX$, for some $X\in 𝔤$ the Lie algebra of $G$. The vector $X$ is called a geodesic vector. The concept can be extended by a minor modification for pseudo-Riemannian homogeneous spaces. Closely related to a g.o. space with reductive decomposition $𝔤=𝔥\oplus 𝔪$, is the concept of a geodesic graph. This is an $\mathrm{Ad}\left(H\right)$-equivariant map $\eta :𝔪\to 𝔥$, which is rational on an open dense subset $U$ of $𝔪$, and such that $X+\eta \left(X\right)$ is a geodesic vector for each $X\in 𝔪$.

The aim of the present paper is to construct certain pseudo-Riemannian g.o. manifolds $M=G/H$. This is done by modifying g.o. metrics that previously appeared in [Z. Dušek, S. Z. Nikčević and O. Kowalski, Differ. Geom Appl. 21, 65–78 (2004; Zbl 1050.22011), C. Gordon, Prog. Nonlinear Differ. Equ. Appl. 20, 155–174 (1996; Zbl 0861.53052) and O. Kowalski and S. Nikčević, Arch. Math. 73, 223–234 (1999; Zbl 0940.53027), Appendix: Arch. Math. 79, 158–160 (2002; Zbl 1029.53063)]. These include the six-dimensional examples $\text{SO}\left(5\right)/\text{U}\left(2\right)$, $\text{SO}\left(4,1\right)/\text{U}\left(2\right)$, and the seven-dimensional examples $\left(\text{SO}\left(5\right)×\text{SO}\left(2\right)\right)/\text{U}\left(2\right)$, $\left(\text{SO}\left(4,1\right)×\text{SO}\left(2\right)\right)/\text{U}\left(2\right)$. For these examples it is shown that the geodesic graphs are nonlinear and discontinuous on an nonempty set, but they are continuous at the origin. The authors conjecture that for every pseudo-Riemannian g.o. space $G/H$ with compact isotropy group $H$, every canonical geodesic graph is continuous at the origin.

##### MSC:
 53C30 Homogeneous manifolds (differential geometry) 53C50 Lorentz manifolds, manifolds with indefinite metrics