The Tao that can be told is not the eternal Tao. The name that can be named is not the eternal name. The nameless is the beginning of heaven and earth. The named is the mother of ten thousand things. Lao Tsu (Tao Te Ching [65], Ch. 1) Mathematics is the music of science and real analysis is the Bach of mathematics. There are many foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. Sterling K. Berberian [11] vii Connections, Sprays and Finsler Structures Downloaded from www.worldscientific.com by 44.224.250.200 on 11/25/20. Re-use and distribution is strictly not permitted, except for Open Access articles. viii Connections, Sprays and Finsler Structures

Abstract. First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of the space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over R n are exactly the Minkowski spaces (Theorem 5 and 6). † * The publication is supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project. The project is co-financed by the European Union and the European Social Fund. †

The space of continuous, SL(m, C)-equivariant, m ≥ 2, and translation covariant valuations taking values in the space of real symmetric tensors on C m ∼ = R 2m of rank r ≥ 0 is completely described. The classification involves the moment tensor valuation for r ≥ 1 and is analogous to the known classification of the corresponding tensor valuations that are SL(2m, R)equivariant, although the method of proof cannot be adapted. * AMS 2010 subject classification: Primary 52B45; Secondary 52A40

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The distance function ̺(p, q) (or d(p, q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over R n , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.