Differential geometry of spray and Finsler spaces (Q2778951)

In chapter 1 the Minkowski spaces and functions are defined and illustrated by several examples. The main scalar of the Cartan torsion is examined and the Varga equation for arbitrary dimension is proved. In chapter 2 the Finsler spaces are defined and the homogeneous Finsler metric is illustrated by the Randers metric, the Berwald-Moór metric, the Poincaré metric, the Ingarden metric, the Lagrange metric, the spherical metric, the Bao-Shen metric, the hyperbolic metric, the Funk metric etc. Relations between some of these metrics are given.NEWLINENEWLINENEWLINEIn chapter 3 the variational problems NEWLINE\[NEWLINE\begin{aligned} \int L\left(x(t), \frac{dx}{dt} (t)\right) dt &= \text{extremum},\\ \int \Phi \left(s,f(s), \frac{df}{ds} (s)\right) ds &= \text{extremum} \end{aligned}NEWLINE\]NEWLINE are examined, which lead to Euler-Lagrange equations. Different metrics lead to different types of these equations. E. Baake's and V. Krivan's model for discounted production in ecology is presented and Matsumoto Finsler metric is mentioned.NEWLINENEWLINENEWLINEIn chapter 4 the spray is defined as a vector field NEWLINE\[NEWLINE G = y^{i} \frac{\partial }{\partial x^{i}} -2G^{i} (y) \frac{\partial }{\partial y^{i}} , G^{i}(\lambda y) = \lambda ^{2}G^{i}(y) , \lambda > 0 . NEWLINE\]NEWLINE A manifold with a spray is called a spray space. A regular curve \(c\) in \(M\) is called a geodesic of \(G\) if it is the projection of an integral curve of \(G\). In several examples the completeness is also examined. The Finsler sprays and geodesics of Funk metrics are presented separately.NEWLINENEWLINENEWLINEIn chapter 5 the different types of volume forms and the \(S\) curvature introduced by Z. Shen are examined, where NEWLINE\[NEWLINE S(y) = \frac{\partial G^{m}}{\partial y^{m}} (y)-\frac{y^{m}}{\sigma (x)} \frac{\partial \sigma }{\partial x^{m}} (x),\quad y = y^{i} \frac{\partial }{\partial x^{i}}|_{x} NEWLINE\]NEWLINE \(d\mu = \sigma (x)dx^{1}\ldots dx^{m}\) is the volume form. It is proved, that every spray on a regular measure space can be deformed to another spray with vanishing \(S\) curvature. The geodesic flow \(\Phi _{t}\) of \(G\) is defined by \(\Phi _{t}(y) = \widehat{c}_{y}(t)\), \(\frac{d\widehat{c}}{dt} = G_{\widehat{c}}\), \(\widehat{c}(0) = y\). Let \(d\mu = \sigma (x) dx^{1}\ldots dx^{n}\) and \(d\widehat{\mu } = \sigma ^{2}(x) dx^{1}\ldots dx^{n}dy^{1}\ldots dy^{n}\) be the volume forms on \(M\) and \(TM\) respectively, and \(\widetilde{G}\) the spray \(\widetilde{G}= G+\frac{2S}{n+1}Y\) where \(Y= Y^i\frac{\partial}{\partial x^i}\), \(Y^k \frac{\partial Y^i} {\partial x^k}+ 2G^i(Y)=0\). The conditions under which the flow \(\widetilde{\Phi}_t\) of \(\widetilde{G}\) preserves the volume form \(\widehat{\mu}\) induced by \(d\mu\) are given. NEWLINENEWLINENEWLINEIn chapter 6 attention is payed to Berwald curvature and its connection with \(S\). Some results of Z. Szabó are mentioned and several applications of Antonelli of two-dimensional conformally Minkowski-Berwald metrics are given. In Finsler surfaces the relations between mean Cartan torsion, Berwald and Landsberg curvature \((L_{ijk} = -\frac{1}{2} g_{ij;k})\) are examined. In chapter 7 different types of connections such as Berwald, Cartan and Chern are studied. Since only Berwald connections can be extended to sprays they play an important role. The geodesics and auto-parallel translations are examined. NEWLINENEWLINENEWLINEThe following important results of the author are given: 1. On any positive definite Berwald space \((M,F)\), the \(S\)-curvature is 0 for the Busemann-Hausdorff volume form \(d\mu _{F}\) and the Holmes Thompson volume form \(d\widetilde{\mu }_{F}\). 2. On a Landsberg space Riemannian tangent spaces are isometric to each other by autoparallel translations along geodesics. NEWLINENEWLINENEWLINEIn chapter 8 the Riemann curvature is introduced by geodesic variation. Flat Ricci-constant and weakly Ricci constant sprays are defined and relations between them and the Ricci scalar are given. The Riemannian curvature of Finsler and Riemann metrics are illustrated by several examples. The \(Y\) related sprays are studied and the theorem of Synge is mentioned. In chapter 9 the structure equations of sprays are examined, further the Bianchi identities and the \(R\)-quadratic and \(R\)-flat sprays. In chapter 10 the Ricci identities of the Finsler metric are presented for \(g\), \(C\) and \(L\) and \(R\)-quadratic and \(R\)-flat Finsler metrics are examined. NEWLINENEWLINENEWLINEIn chapter 11 the Finsler spaces of scalar and constant curvature are studied and illustrated by several examples. In chapter 12 are mentioned several theorems which give conditions for two Finsler metrics to be pointwise projective (Rapcsák) and their connection with sprays. The inverse problem is also studied, namely, when the given spray is locally or globally Finslerian. NEWLINENEWLINENEWLINEIn chapter 13 different curvatures of sprays, such as Douglas curvature, Weyl curvature, Berwald-Weyl curvature are examined. Isotropic and projectively affine sprays are studied. In chapter 14 a numbers of new and important results connected with exponential map, Jacobi fields and sprays are given. Many of them are original. NEWLINENEWLINENEWLINEIn the book the author presents numerous results connected with Finsler spaces and spray theory obtained during the last hundred years. It was difficult to unify the notations and the terminology. Many old results are rediscovered and several results of the author are composed in the book, so that it all gives a unique construction. Numerous examples make the reading of the book easier and more interesting. As the theory of sprays gains more and more application, this book appeared just in time to fill up the gap on this subject.