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Applications of reducible and semireducible metric spaces of linear elements to the theory of motions. (Russian) Zbl 0841.53017

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.

MSC:

53B20 Local Riemannian geometry
54H15 Transformation groups and semigroups (topological aspects)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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