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Metric Characterizations of Projective-Metric Spaces

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Abstract

This chapter is concerned with the study of projective-metric spaces, that is, metrics on open subsets of projective space whose geodesics are the intersection of this open set with the lines of the ambient space. The stress is on the effect of additional conditions on these so-called “projective-metric spaces”, which lead to some characterization of special geometries. We rely heavily on the work of Herbert Busemann in this domain. We formulate many open problems on this subject.

The research leading to these results has received funding from the national projects TKP2021-NVA-09 and NKFIH-1279-2/2020. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme. Project no. NKFIH-1279-2/2020 has been implemented with the support provided by the Ministry for Innovation and Technology of Hungary (MITH) under grant NKFIH-1279-2/2020.

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Notes

  1. 1.

    Some authors use the weaker condition that the segments are geodesics that makes every Hilbert metric a projective-metric space.

  2. 2.

    If \(\lambda _{\boldsymbol {v}}^\pm =\infty \), then \(P_{\boldsymbol {v}}^\pm \) is an ideal point.

  3. 3.

    This means that Busemann convexity [75, Definition 8.1.1] is not satisfied in this space.

  4. 4.

    To investigate regular polygons in projective-metric spaces was an idea of Gábor Korchmáros raised amid an oral conversation in 2018.

  5. 5.

    Notice however that the proof does not need, instead implies the parallel axiom.

  6. 6.

    It seems well-founded in two manuscripts [59, 62] of the author.

  7. 7.

    A Riemannian point of a Hilbert plane is where the tangent plane is Riemannian.

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Acknowledgements

It is a pleasure to thank Athanase Papadopoulos for inviting me to contribute a chapter to Surveys in Geometry Vol. II.

I thank József Kozma, and Athanase Papadopoulos for their help.

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    Árpád Kurusa

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    Athanase Papadopoulos

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Kurusa, Á. (2024). Metric Characterizations of Projective-Metric Spaces. In: Papadopoulos, A. (eds) Surveys in Geometry II. Springer, Cham. https://doi.org/10.1007/978-3-031-43510-2_7

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