By Demeter Krupka, David Saunders

This can be a accomplished exposition of subject matters lined by way of the yank Mathematical Society's category "Global Analysis", facing smooth advancements in calculus expressed utilizing summary terminology. it will likely be helpful for graduate scholars and researchers embarking on complicated reports in arithmetic and mathematical physics. This booklet presents a complete assurance of recent international research and geometrical mathematical physics, facing subject matters corresponding to; constructions on manifolds, pseudogroups, Lie groupoids, and worldwide Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential structures and distributions, and spectral theory); variational idea on manifolds, with functions to physics; functionality areas on manifolds; jets, usual bundles and generalizations; and non-commutative geometry. - finished insurance of contemporary worldwide research and geometrical mathematical physics - Written through world-experts within the box - updated contents

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2003) 491–509 J. M. Lee: Riemannian manifolds – An Introduction to Curvature (Springer, 1997) Yiming Long: Multiplicity and stability of closed geodesics on Finsler 2-spheres J. Eur. Math. Soc. (JEMS) 8 (2), (2006) 341–353 M. Matsumoto: Foundations of Finsler Geometry and Special Finsler Spaces (Japan: Kaiseisha-Press, 1986) S. Mazur and S. Ulam: Sur les transformations isom´etriques d’espaces vectoriels norm´es C. R. Acad. Sci. Paris 194 (1932) 946–948 X. Mo: An Introduction to Finsler Geometry (Peking University Series in Mathematics, Vol.

1 Closed geodesics The study on closed geodesics on spheres is a classical and important problem in both dynamical systems and differential geometry. The results of V. Bangert in 1993 and J. Franks in 1992 prove that for every Riemannian metric on S 2 there exist infinitely many geometrically distinct closed geodesics. In contrast, in 1973, A. Katok ([24]) constructed a 34 Global aspects of Finsler geometry remarkable irreversible Finsler metric on S 2 which possesses precisely two distinct prime closed geodesics.

7 Let (M, F ) be a connected Berwald surface with smoth and strongly convex F on T M \ 0. Then • if the Gauss curvature K vanishes identically, then F is locally Minkowskian, • if the Gauss curvature K is not identically zero, then F is Riemannian. The proof is based on the following fact: If (M, F ) is a Landsberg surface with smooth F on T M \ 0, then the value of the Gauss curvature K at any point of the indicatric Sx M Tadashi Aikou and L´aszl´o Kozma 33 is determined by the Cartan scalar I accoding to the following formula: K(t) = K(0)e t 0 I(τ ) dτ .