Metric characterizations of projective-metric spaces. (English) Zbl 07918844
Papadopoulos, Athanase (ed.), Surveys in geometry II. Cham: Springer. 225-256 (2024).
Summary: 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.
For the entire collection see [Zbl 1537.51001].
For the entire collection see [Zbl 1537.51001].
MSC:
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |
| 51F10 | Absolute spaces in metric geometry |
| 52A01 | Axiomatic and generalized convexity |
| 51B20 | Minkowski geometries in nonlinear incidence geometry |
| 53A40 | Other special differential geometries |
Keywords:
metric characterizations; projective-metric spaces; Minkowski geometry; Hilbert geometry; Busemann spaces; constant curvature spaces; ellipses; hyperbolas; conicsReferences:
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