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A setting for spray and Finsler geometry. (English) Zbl 1105.53043
Antonelli, Peter L. (ed.), Handbook of Finsler geometry. Vols. 1 and 2. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1557-7/set; 1-4020-1555-0/v.1; 1-4020-1556-9/v.2). 1183-1426 (2003).
This part of the Handbook gives a systematic and fairly comprehensive account of the fundamentals of a manifold endowed with second-order differential equations or, in particular, a Finsler metric. The framework for this is provided by the vector bundles. The theory presented is formulated in the pull-back vector bundle $$\tau^*_M\tau M=(TM \times_M TM,p_1,TM)$$. The main tools are the Frölicher-Nijenhuis calculus as applied to semibasic scalar- and vector-valued forms over $$TM$$, and covariant derivations in $$\tau^*_M\tau M$$. In most cases a Berwald derivative coming from a nonlinear connection is applied. Proceeding these lines this work seems to be the first which is self-contained. Throughout the work the author rigorously insists on the clear concepts set up at the beginning. The presentation, starting from the fundamental principles, is of a high level, yet it is well understandable.
Chapter 1 contains the fundamentals of manifolds, vector bundles, covariant derivation operators in vector bundles, basic differential operators on a manifold and the notions to be used later. Some indispensable results will also be stated here, so that the reader will not be forced to look for them in other books. Proofs are generally omitted in this chapter.
Chapter 2 forms the backbone of the study. Here the general constructions presented in Chapter 1 are applied to the pull-back bundle $$\tau^*_M\tau M$$. In this chapter a fairly comprehensive theory of nonlinear connections in vector bundles, the Frölicher-Nijenhuis theory and the elements of the Martinez-Carenina-Sarlet theory are worked out. Covariant derivative operators in $$\tau^*_M\tau M$$, and in particular the Berwald derivative are discussed in this context. In this and in the next chapters most of the results are proved in detail.
Chapter 3 demonstrates how the general theory works in practice. It is applied to a geometric treatment of second-order differential equations and to a deduction of the theorems providing the foundation of the theory of Finsler manifolds. These appear in the main text as a second-order vector field. Semisprays and sprays are related notions. Any second-order vector field (semispray, spray) generates a nonlinear connection, and hence a Berwald derivative in $$\tau^*_M\tau M$$. This Berwald derivative is an efficient tool in the study of the linearizability properties of a second-order vector field. The theory of Finsler manifolds is unconventionally built up. Instead of a Lagrange function or an energy function the author starts with a metric tensor which leads to a Finsler manifold if it satisfies the normality condition of M. Hashiguchi. Also the regularity condition of R. Miron plays an important role.
The Appendix is actually a supplement to Chapter 1. It provides a glossary of notational conventions, set-theoretical and topological concepts, and summarizes the necessary algebraic background.
To comprehend this work no prior knowledge on connection theory of Finsler geometry is required. This is a self-contained treatment of the subject immediately accessible for graduate students or interested physicists and biologists.
For the entire collection see [Zbl 1057.53001].

##### MSC:
 53C40 Global submanifolds 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
##### Keywords:
Finsler metric; second order ODE; sprays