Introduction to Finsler geometry. Paper from the 29th Brazilian mathematics colloquium -- 29\(^{\text o}\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 22 -- August 2, 2013 (Q2848163)

This small book (200 pages) provides a complete and self sufficient brief introduction to Finsler geometry including the study of some dynamical aspects of its geodesic flow, in particular some applications of modern symplectic dynamics.NEWLINENEWLINENEWLINEIn the first chapter the authors introduce the notion of Finsler metric and explain some examples; some classical ones (Riemannian metric and Finsler metrics of Randers type, Zermelo type and (\(\alpha,\beta\)) type), but also non-classical ones (Katok, Shen and Matsumoto metrics).NEWLINENEWLINEIn the second chapter (about connections and curvature), the authors study geometric structures associated to Finsler metrics. They follow J. Grifone's formalism. This chapter includes, among other topics, sections on horizontal distributions and Grifone connections, sprays, Berwald connections, Cartan connections, Chern-Rund connections, Hashigushi connections, symmetric Finsler connections, regular lifts of Grifone connections, covariant derivative and parallel transport, Riemannian and flag curvatures.NEWLINENEWLINEIn the third chapter, the authors deal with Morse theory and applications. The chapter includes sections on basic Morse theory, Finsler geometry, covariant derivative along a geodesic, Jacobi fields and geodesic flow, exponential map, set of paths, first and second variations of energy, Morse theory for the energy functional and applications (geodesics in spheres, Auslander's theorem, Finsler version of Cartan-Hadamard theorem, Rademacher's sphere theorem). The authors make the reader aware of the differences between the Riemannian and the Finsler setting regarding this topic.NEWLINENEWLINEIn the forth chapter they introduce symplectic and contact manifolds and explain some connections with Finsler geometry in both types of manifolds.NEWLINENEWLINEIn the last chapter the authors study the geodesic flow from the symplectic dynamic point of view. This chapter includes three major sections: Conley-Zehnder index, dynamic convexity and Poincaré-Birkhoff theorem.