×

zbMATH — the first resource for mathematics

On the generalized Lagrange spaces with the metric \(\gamma_{ij}(x)+(1/c^ 2)y_ iy_ j\). (English) Zbl 0703.53021
In the generalized Lagrange space \(M^ n\) the special metric \(g_{ij}(x,y)=\gamma_{ij}(x)+c^{-2}y_ iy_ j\) is considered, where \(y_ i=\gamma_{ij}(x)y^ j.\) Using the absolute energy \(\epsilon (x,y)=g_{ij}(x,y)y^ iy^ j\) a new metric \(g^*_{ij}(x,y)=2^{-1}\frac{\partial^ 2\epsilon}{\partial y^ i\partial y^ j}\) is introduced. The nonlinear connection \(N_ j^{*i}=\partial G^{*i}/\partial y^ j\) is globally defined. With respect to this \(N^*\) the adapted basis is given and the metric connection coefficients, the torsion and curvature tensors are determined. The equations of the horizontal and vertical geodesic lines and paths are given. The Einstein equation with respect to \(g_{ij}\) are: \[ R_{ij}-2^{-1}Rg_{ij}=\chi T^ H_{ij},\quad S_{ij}- 2^{-1}Sg_{ij}=\chi T^ V_{ij}, \] where T is the energy momentum tensor. The special cases, as locally Minkowski space, are examined and the Hermitian model of the space \(M^ n\) is given.
Reviewer: I.Comic

MSC:
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
PDF BibTeX XML Cite