Egorov, A. I.; Egorova, L. I. Applications of reducible and semireducible metric spaces of linear elements to the theory of motions. (Russian) Zbl 0841.53017 Tr. Geom. Semin. 22, 12-31 (1994). Let us denote by \({\mathop B \limits^m}_n\) the metric space of linear elements, i.e. a manifold of linear elements \(X_{2n-1}(x,y)\) supplied with a tensor field \(b_{\alpha\beta}(x,y)\) such that \(\text{det}|b_{\alpha\beta} (x,y)|\neq 0\) and \(b_{\alpha \beta} (x,\lambda y) = \lambda^m b_{\alpha\beta} (x,y)\) (\({\mathop B \limits^0}_n\) is a Finsler space if \(b_{\alpha\beta} = b_{\beta\alpha}\) and \(\partial_\alpha b_{\beta\gamma} = \partial_\beta b_{\alpha\gamma}\)). \({\mathop B \limits^m}_n\) is called a space of \(k\)-th lacunarity if the dimension \(r\) of its motion group \(G_r\) lies in the \(k\)-th condensation segment.In the paper there are found necessary and sufficient conditions for \({\mathop B \limits^m}_n\) with positive definite metric of the adjoint space to be a space of \(k+1\)-th lacunarity with maximal dimension \(r\) of \(G_r\) \(\biggl(r = {(n-k)(n-k+1)\over 2} + {k(k+1) \over 2}\biggr)\), and the cases \(k = 0\) and \(k = 1\) are studied in detail. The authors define reducible and semireducible metric spaces of linear elements (analogues of Riemannian spaces) and prove that each \({\mathop B \limits^m}_n\) of \(k\)-th lacunarity whose \(G_r\) has the maximal dimension, is semireducible. Reviewer: M.Malakhal’tsev (Kazan’) MSC: 53B20 Local Riemannian geometry 54H15 Transformation groups and semigroups (topological aspects) 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics) Keywords:reducible space; lacuna; metric space of linear elements; Finsler space; motion group PDFBibTeX XMLCite \textit{A. I. Egorov} and \textit{L. I. Egorova}, Tr. Geom. Semin. 22, 12--31 (1994; Zbl 0841.53017) Full Text: EuDML