Geometriák karakterizálása projektív-metrikus terekben

Abstract

In the dissertation, we present our research in the fields of projective metric geometries, in the course of which we characterize the hyperbolic and Euclidean geometry among Hilbert, respectively Minkowski geometries by geometric configurations. Most mathematicians know Minkowski geometries as normed vector spaces, and a wide literature counts their attributes and properties. The research of Hilbert geometry falls into line in our days, with further various results. Likewise Minkowski geometry is a straight generalisation of the Euclidean geometry, an immediate generalisation of hyperbolic geometry is Hilbert geometry. That is why investigations of properties of configurations well known from Euclidean and hyperbolic geometries is of prime importance in exploring these geometries. Therefore, the general starting point of our dissertation was the question: what kind of properties do have some configurations - deliberately characterized in other geometries - in Minkowski and Hilbert geometries, furthermore, in case of fulfilling certain conditions, what sort of consequences shall we have take into account? Pursuant to this, a survey of the basic features of geometries under investigation is given. Beyond that of Minkowski geometries, a description of the hyperbolic and Hilbert geometries is given, and afterwards the following questions are investigated: - What properties of the particular configurations continue in the general case, and which properties will stop to characterize the configuration? - What kind of consequences implies for the whole geometry if a certain property of a configuration is required to retain? - Are there any configuration and a particular property of it which block the way of generalisation (i.e. the geometry will be Euclidean, respectively, hyperbolic)? In case of all questions worth to bring up the following problem: if the fulfilment of a property is not a general requirement, but holds only for some specific cases, then will it result in the same result? The dissertation draws a picture of investigations which lead, without exception, to characterisation of classic Euclidean, respectively hyperbolic geometries among Minkowski, respectively Hilbert geometries. We start with investigation of some significant configurations in connection with triangles, analogous theorems about which are well known in hyperbolic geometry, as well. A proof, in the Cayley-Klein model, is shown for the hyperbolic version of the Ceva's and Menelaus' theorems (Hyperbolic Menelaus' theorem and Hyperbolic Ceva's theorem) which automatically come true in Minkowski geometries, as well as for the statement that altitudes, respectively orthogonal bisectors of triangles belong to a bundle (Theorem on hyperbolic orthocentre and Theorem on hyperbolic bisectoral centre). As the Birkhoff-orthogonality is not symmetric, concurrency of perpendicular bisectors and that of altitudes should be treated separately in Minkowski and Hibert geometries in the case of Birkhoff-perpendicularity (left-perpendicularity for later use), and in the case of its inverse relation, called right-perpendicularity, or H-perpendicularity, for the sake of distinction. In the course of our research the inscribed (maximal volume) respectively the circumscribed (minimal volume) ellipsoids (called Loewner, repsectively John ellipsoids) of strictly convex bodies play a prominent role. Some basic statements regarding the tangent points of these ellipsoids to the convex bodies are considered. Besides several statements of technical kind, we prove a result about ellipse characterisation, and a theorem of Ceva type about inscribed triangles of ovals, which are interesting on their own, as well. The ellipse characterisation build upon harmonic division, dual of which is equivalent to - by means of Ceva's and Menelaus' theorem - a theorem of Segre. With respect to Hilbert geometries is proven that hiperbolic geometry is characterized by the property that every trigon possesses - the Ceva property; - the Menelaus property; - a bisectoral centre (i.e., perpendicular bisectors belong to a bundle); - an orthocentre {i.e., altitudes belong to a bundle). As regards latter two statements, we have to mention that they apply the inverse of the Birkhoff-perpendicularity, called H-perpendicularity, as we could not achieve any result in the case of Birkhoff-perpendicularity, and we could not find any reference to a result of that kind in the literature. In Minkowski spaces, it is proven equally for the case of the left- and the right-perpendicularity that the Euclidean geometry is characterised by the property that every trigon possesses - the right-bisectoral centre; - the right-orthocentre; - the left-bisectoral centre; - the left-orthocentre...