Abstract
Two homogeneous pseudo-Riemannian manifolds \((G/H, \textrm{d}s^2)\) and \((G'/H', \textrm{d}s'^2)\) belong to the same real form family if their complexifications \((G_\mathbb {C}/H_\mathbb {C}, \textrm{d}s_\mathbb {C}^2)\) and \((G_\mathbb {C}'/H_\mathbb {C}', \textrm{d}s_\mathbb {C}'^2)\) are isometric. The point is that in many cases, a particular space \((G/H, \textrm{d}s^2)\) has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if \((G/H, \textrm{d}s^2)\) is a geodesic orbit space with a reductive decomposition \(\mathfrak {g}= \mathfrak {h}\oplus \mathfrak {m}\), then the same holds all the members of its real form family. In particular,our understanding of compact geodesic orbit Riemannian manifolds gives information on geodesic orbit pseudo-Riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D’Atri spaces, and a number of open problems.
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Notes
By an abuse of notation, we write \(\textrm{Ad}(H)\) instead of \(S(\textrm{Ad}(H))\) for the symmetric powers that form the action of H on \(S(\mathfrak {m})\).
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Acknowledgements
Research partially supported by a Simons Foundation Grant.
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Wolf, J.A. Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds. J Geom Anal 33, 214 (2023). https://doi.org/10.1007/s12220-023-01250-9
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DOI: https://doi.org/10.1007/s12220-023-01250-9
Keywords
- Geodesic orbit space
- Real form family
- Weakly symmetric space
- Naturally reductive space
- Commutative space
- D’Atri space
- Pseudo-Riemannian manifold
- Homogeneous manifold