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Family of subspaces in a Finsler space. II. (English) Zbl 0632.53027
[Part I, cf. Tensor, New Ser. 37, 187-197 (1982; Zbl 0489.53030).]
In the Finsler space \(F_ n\) two families of linearly independent and mutually normal vector fields \(B_ a^{\alpha}(x,\dot x)\), \(a=1,2,...,m\), and \(N_ k^{\alpha}(x,\dot x)\), \(k=m+1,...,n\), are given. The vectors \(dx^{\alpha}\), \(\dot x^{\alpha}\), \(DB_ a^{\alpha}\), \(DN_ k^{\alpha}\), \(D\ell^{\alpha}\) are decomposed in the direction of these two families. If \(B_ a^{\alpha}=B_ a^{\alpha}(x)\) and \(\partial_ bB_ a^{\alpha}=\partial_ aB_ b^{\alpha}\) then the vector fields \(B_ a^{\alpha}(x)\) determine a family of subspaces for which they are tangent vectors and for which the \(N_ k^{\alpha}\) are the normal vectors. For this special case the coefficients of the induced connection are obtained which depend on some parameters. Conditions for the parameters are given such that the induced connection be metric. [\(\Delta\) D] \(B_ a^{\alpha}\) and [\(\Delta\) D] \(N_ k^{\alpha}\) are expressed by the induced curvature tensors.
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
Zbl 0489.53030