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
We establish the formulas on the power τ k of the tangent bundle τ = τ (RP n) of the real projective n-space RP n and its complexification cτ k , and apply the formulas to the problem of extendibility and stable extendiblity of τ k and cτ k .
After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-free covariant derivative on the base manifold compatible with the Finsler function and Aikou's characterization of Berwald manifolds. Finally, we study some implications of V. Matveev's observation according to which quadratic convexity may be omitted from the definition of a Berwald manifold. These include, among others, a generalization of Z. I. Szabó's wellknown metrization theorem, and leads also to a natural generalization of Berwald manifolds, to Berwald -Matveev manifolds.
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