Finsler metrics compatible with a special Riemannian structure (Q2706511)

The paper contains results about the structure \(f^i_j\neq 0\) satisfying NEWLINE\[NEWLINEf^i_rf^r_kf^k_hf^h_lf^l_tf^t_j+f^i_rf^r_j=0,\text{ rank }(f^i_r)= 2r\leq n\tag{1}NEWLINE\]NEWLINE in a Finsler space \(F^n\), whose metric is compatible with \(f^i_j\). The complementary projection operators \({\mathbf l}^i_j=-f^i_rf^r_k f^k_hf^h_j\), \({\mathbf m}^i_j=f^i_r f^r_kf^k_hf^h_j+ \delta^i_j\) on the tangent space \(T_p(F^n)\) at any point \(p\) of the Finsler space \(F^n\) with metric \(L(x,y)\) define distributions \({\mathcal L}\) and \({\mathcal M}\). \((L(x,y)=\|y \|.)\) Since \(f^i_rf^r_j\) is an almost complex structure on the subspace \({\mathcal L}\) in \(T_p(F^n)\), the product of a complex number \(\overline c=|\overline c|(\cos \theta+i\sin \theta)\) with any \({\mathbf l}^i_ry^r\) on \({\mathcal L}\) is defined as follows: \(\overline c{\mathbf l}^i_ry^r= |\overline c|(\delta^i_j \cos\theta+ f^i_rf^r_j\sin \theta){\mathbf l}^j_ky^k\). For \(f_\theta\) defined by \(f^i_{\theta j}= \delta^i_j \cos\theta+ f^i_rf^r_j \sin\theta\) holds \(L(x,f_\theta {\mathbf l}y)= L(x,{\mathbf l}y)\), i.e. \(f^i_j\) is compatible with \(L\). Such \(F^n\) is called an \((f^2,L)\)-manifold. Further, conditions equivalent to this are given. Is an \((f^2,L)\)-manifold the distribution \({\mathcal L}\) is a complex manifold, if the \(h\)-covariant derivate \(\nabla_k\) of the structure \(f^i_j\) satisfying (1) with respect to the Cartan connection vanishes. The relation between \(\nabla_k\), and \(h\)-covariant derivate \(\overline\nabla_k\) with respect to the Berwald connection are studied. Obtained are conditions under which the \(h\)-curvature tensor of the Berwald connection vanishes and conditions for \(f^i_j\), \(\nabla_k\) which have to be fulfilled so that the distribution \({\mathcal L}\) be a Berwald space.