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On Non-positive Curvature Properties of the Hilbert Metric

The Journal of Geometric Analysis volume 29, pages 569–576 (2019)Cite this article

Abstract

In this paper, we consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in \({\mathbb {R}^n}\). First, we survey the relationships among the concepts and prove that in the case of Hilbert metric some of them are equivalent. Furthermore, we show some condition which implies the rigidity feature: if the Hilbert metric is Berwald, i.e., its Finslerian Chern connection reduces to a linear one, then the domain is an ellipsoid and the metric is Riemannian.

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  1. Institute of Mathematics, University of Debrecen, 4002, Debrecen, P.O. Box 400, Hungary

    Layth M. Alabdulsada & László Kozma

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  1. Layth M. Alabdulsada
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  2. László Kozma
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Correspondence to László Kozma.

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Alabdulsada, L.M., Kozma, L. On Non-positive Curvature Properties of the Hilbert Metric. J Geom Anal 29, 569–576 (2019). https://doi.org/10.1007/s12220-018-0011-9

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  • DOI: https://doi.org/10.1007/s12220-018-0011-9

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