Invariant frames for a generalized Lagrange space with Berwald-Moor metric
2005jbm | Marius Paun  // Faculty of Mathematics and Informatics, Transilvania, University of Brasov, Romania

The notion of generalized Lagrange space should be geometrically considered as a generalized metric space $M^n=(M,g_{ij}(x,y))$. A theory of invariant Finsler spaces was given by M. Matsumoto and R. Miron with important applications. The notion of non-holonomic space was introduced by Gh. Vranceanu in [VR]. The Vranceanu type invariant frames and the invariant geometry of second order Lagrange spaces was studied by the author in [P3]. The purpose of the present paper is to study the invariant geometry for a generalized Lagrange space endowed with a Berwald-Moor metric. We introduce distinct non-holonomic frames on the two components of the Whitney's decomposition. This will determine a non-holonomic coordinates system on the total space $TM$ and thus its geometry can be studied with methods analogous to the mobile frame. We obtain, in this manner, invariant connections, curvatures and torsions, and the fundamental equations in this theory. Also we can construct the invariant frames so that, with respect to them, the metric of the total space can be written in canonical form and in this case we deduce invariant Einstein equations. We mention that the frames introduced here depend on the metric and all the computations are for this metric.

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