Abstract:In the 1980s, M. J. Markowitz introduced a conformally invariant pseudodistance on pseudo-Riemannian manifolds, inspired by the Kobayashi metric in projective geometry. This construction relies on a distinguished class of parametrized lightlike geodesics, called projectively parametrized. We begin by reviewing the fundamental properties of this pseudodistance and provide several families of examples where it is non-degenerate and, in some cases, complete. In particular, we investigate three classes of manifolds: closed manifolds, conformally convex domains of the Einstein universe, and globally hyperbolic, conformally flat, $C$-maximal spacetimes. For the first two classes, we obtain results analogous to those of Brody and Barth concerning the complex Kobayashi metric. Finally, we apply Markowitz's pseudodistance to classify all quasi-homogeneous domains of the Einstein-de Sitter space, that is, a half-space of Minkowski space bounded by a spacelike hyperplane. Up to conformal transformations, only finitely many such domains exist, and all of them turn out to be homogeneous.
| Comments: | Minor corrections. Final version. To appear in Geometriae Dedicata |
| Subjects: | Differential Geometry (math.DG); Mathematical Physics (math-ph); Metric Geometry (math.MG) |
| Cite as: | arXiv:2509.15745 [math.DG] |
| (or arXiv:2509.15745v2 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2509.15745 arXiv-issued DOI via DataCite |
From: Adam Chalumeau [view email] [via CCSD proxy]
[v1]
Fri, 19 Sep 2025 08:15:05 UTC (49 KB)
[v2]
Fri, 22 May 2026 14:38:29 UTC (51 KB)
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