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The geometry of Lagrange spaces: theory and applications. (English) Zbl 0831.53001
Fundamental Theories of Physics. 59. Dordrecht: Kluwer Academic Publishers. xiv, 285 p. (1994).
Chapter I is devoted to very general notion of fibre bundle. The particular cases of principal fibre bundles and vector bundles are discussed. The concept of nonlinear connection \(N\) in the vector bundle \(\xi = (E, p, M)\) is defined and the local representation of the nonlinear connection \(N\) is given by the real function \(N^a_i(x,y)\) defined on \(p^{-1} (u)\). The nonlinear connection in the bundle \((E - 0, p|_{E - 0}, M)\) is considered, which is essential in Finsler geometry. The geometry of the total space \(E\) is developed by introducing the concept of \(d\)-linear connection on \(E\). That is a linear connection \(D\) which preserves by parallelism the horizontal and vertical distribution. Structure equations of a \(d\)-linear connection are given, using the \(h\)- and \(v\)-covariant derivations determined by the \(d\)- connection \(D\). If \(G\) is a metrical structure on \(E\), then there exists a nonlinear canonical connection derived from \(G\). The metrical \(d\)- connection is introduced. The idea of using this nonlinear canonically connection was proposed by R. Miron.
The geometrical theory of embeddings of vector bundles is extended in Chapter IV, using the method given by R. Miron in the theory of subspaces of Lagrange spaces. The Gauss-Weingarten formulae and the Gauss-Codazzi equations are derived. Using the canonical metrical \(d\)-connection on \((E, G)\), in Chapter V are obtained the Einstein equations associated to it as well as the conservation laws. The case when the fibre dimension of \(\xi = (E, p, M)\) is one is treated.
In Chapter VI, a gauge covariant derivative is defined. Using a gauge metrical \(d\)-connection the Einstein-Yang-Mills equations are generalized. In Chapter VII, the geometry of the total space \(E = TM\) of the tangent bundle \((TM, \tau, M)\) is obtained by particularising the geometry of \(E\). Also studied are: the almost complex structure \(F\) on \(TM\), derived from a nonlinear connection the class of \(d\)-connections which are compatible with \(F\), and the metrical \(d\)-connections on \(TM\). Some remarkable metrics on \(TM\) are analysed. The Finsler space, the Lagrange spaces and the generalized Lagrange spaces are studied by a unitary theory in Chapter VIII - X. This generalisation is due to the first author.
The Lagrange geometry is extensively presented in this book. New methods for constructing geometrical models of theoretical mechanics and gravitational and electromagnetic fields are obtained. A simple idea of R. Miron was to consider the Einstein equations in Lagrange space as the equations associated to the canonical metrical connections from an almost Hermitian model.
The Finsler and Lagrange spaces are examples of generalized Lagrange spaces, but there exist generalised Lagrange spaces which are not reducible to Finsler or Lagrange spaces. The geometry of generalized Lagrange spaces is developed in Chapter X. Using this theory, in Chapter XII a new geometrical model for relativistic optics of dispersive media is obtained. In Chapter XIII a geometrization of time dependent Lagrangians is presented, using the theory of non-linear connections in \(\xi = (\mathbb{R} \times TM, \pi, \mathbb{R} \times M)\) and the metrical \(d\)- connection on \(\mathbb{R} \times TM\). A metrical almost contact model of a rheonomic Lagrange space is given. One obtains generalized rheonomic Lagrange spaces.
This book is an excellent monograph. The style is elegant and compact.
Reviewer: P.Stavre (Craiova)

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53Z05 Applications of differential geometry to physics