By Demeter Krupka, David Saunders

ISBN-10: 0080556736

ISBN-13: 9780080556734

ISBN-10: 0444528334

ISBN-13: 9780444528339

This can be a accomplished exposition of issues coated through the yank Mathematical Society's category "Global Analysis", facing smooth advancements in calculus expressed utilizing summary terminology. it will likely be priceless for graduate scholars and researchers embarking on complicated experiences in arithmetic and mathematical physics. This booklet offers a entire assurance of recent international research and geometrical mathematical physics, facing themes corresponding to; buildings on manifolds, pseudogroups, Lie groupoids, and international Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential platforms and distributions, and spectral theory); variational concept on manifolds, with functions to physics; functionality areas on manifolds; jets, ordinary bundles and generalizations; and non-commutative geometry. - complete assurance of recent international research and geometrical mathematical physics - Written via world-experts within the box - updated contents

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3 Let (M, F ) be a closed Finsler manifold with negative flag curvature. If F has constant S-curvature, then it must be Riemannian. Let K(P, y) be the flag curvature of the Finsler space (M, F ). One calls (M, F ) a Finsler space of scalar curvature if K(P, y) = K(x, y) is independent of the flags P (associated with any fixed flagpole y) Mo and Shen ([35]) proved that, for any n-dimensional (n ≥ 3) compact negatively curved Finsler spaces (M, F ) of scalar curvature, F is a Randers metric. 4 Let (M, F ) be an n-dimensional complete Finsler manifold of scalar flag curvature K = K(x, y) ≤ −1 (n ≥ 3).

1 Curvature rigidity Rigidity results state that under such and such assumptions about the curvature, the underlying Finsler structure must be either Riemannian or locally Minkowskian. A famous result of this type is Akbar-Zadeh’s theorem ([2]) about spaces with constant flag curvature (a generalization of the sectional curvature). 1 ([2]) Let (M, F ) be a compact connected boundaryless Finsler manifold of constant flag curvature λ. • If λ < 0, then (M, F ) is Riemannian. • If λ = 0, then (M, F ) is locally Minkowskian.

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. See a fine analysis about it in [52]. Closed geodesics on a compact manifold with a Finsler metric F can be characterized as the critical points of the energy functional E : ΛM → R; E(γ) = 1 2 1 F 2 (γ (t)) dt.

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