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On non-positive curvature properties of the Hilbert metric. (English) Zbl 1411.53057

The authors of the paper consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in \(\mathbb{R}^n\). Further they show some condition which implies the rigidity feature: If the Hilbert metric is Berwald then the domain is an ellipsoid and the metric is Riemannian.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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References:

[1] Bačák, M., Hua, B., Jost, J., Kell, M., Schikorra, A.: A notion of nonpositive curvature for general metric spaces. Differ. Geom. Appl. 38, 22-32 (2015) · Zbl 1322.53040 · doi:10.1016/j.difgeo.2014.11.002
[2] Bao, D., Chern, S.S., Shen, Z.: Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics. Springer, New York (2000) · Zbl 0954.53001 · doi:10.1007/978-1-4612-1268-3
[3] Busemann, H.: The Geometry of Geodesics. Academic Press, New York (1955) · Zbl 0112.37002
[4] Gu, S.: On the equivalence of Alexandrov curvature and Busemann curvature. Turk. J. Math. 41(1), 211-215 (2017) · Zbl 1424.53078 · doi:10.3906/mat-1509-52
[5] Guo, R.: Characterizations of hyperbolic geometry among Hilbert geometries: a survey. Handbook of Hilbert Geometry, pp. 147-158. IRMA Lect. Math. Theor. Phys., 22, Eur. Math. Soc., Zürich (2014)
[6] Jost, J.: Nonpositivity Curvature: Geometric and Analytic Aspects. Lecture in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1997) · Zbl 0896.53002
[7] Kelly, P., Straus, E.: Curvature in Hilbert geometries. Pac. J. Math. 8, 119-125 (1958) · Zbl 0081.16401 · doi:10.2140/pjm.1958.8.119
[8] Kristály, A., Kozma, L.: Metric characterization of Berwald spaces of non-positive flag curvature. J. Geom. Phys. 56(8), 1257-1270 (2006) · Zbl 1103.53046 · doi:10.1016/j.geomphys.2005.06.014
[9] Okada, T.: On models of projectively flat Finsler spaces of constant negative curvature. Tensor (N.S.) 40(2), 117-124 (1983) · Zbl 0558.53022
[10] Papadopoulos, A.: Metric Spaces, Convexity and Non-positive Curvature, 2nd edn. IRMA Lectures in Mathematics and Theoretical Physics, 6. European Mathematical Society (EMS), Zürich (2014) · Zbl 1296.53007
[11] Troyanov, M.: Funk and Hilbert geometries from the Finslerian viewpoint, Handbook of Hilbert geometry, pp. 69 -110. IRMA Lect. Math. Theor. Phys., 22. Eur. Math. Soc., Zürich (2014)
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