On Holland's frame for Randers space and its applications in physics (Q2733926)

If \(\{{\delta\over \delta x^i}, {\partial\over \partial y^i}\}\) is the adapted basis in the second tangent bundle TTM of a Finsler space, then the anholonomic Finsler frame is defined by NEWLINE\[NEWLINE\left\{{\delta \over\delta x^\alpha} =y^i_\alpha {\delta\over \delta x^i},\;{\partial\over \partial y^\alpha} =y^i_\alpha {\partial\over \partial y^i}\right\}, \quad\alpha=1, \dots, n,NEWLINE\]NEWLINE where \([Y^i_\alpha]\) is a regular matrix and \([Y^\alpha_i]\) its inverse. The relations between \(d\)-connection, curvature and torsion tensors in adapted and anholonomic frames are given. The frame \(\{Y^i_\alpha\}\) is defined as holonomic if there are functions \(\varphi^\alpha\), such that \(Y^\alpha_i dx^i =d \varphi^\alpha\). The holonomicity conditions are obtained. For a given anholonomic Finsler frame \(\{Y^i_\alpha\}\), there exists a unique \(d\)-connection \(\Delta\), for which \(Y^i_\alpha,j=0\) and \(Y^i_\alpha |_j=0\). This connection is the crystallographic connection and has the property, that all its curvature tensors are equal to zero. For the Randers metric \(L(x,y)=\alpha (x,y)+b_i(x) y^i\) the anholonomic Holland's frame is constructed. For this frame is \(g_{ij}= Y^\alpha_iY^\beta_j g_{\alpha,\beta}\). A Randers space with Holland's frame and crystallographic connection is flat. Some other properties are proved.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].