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Adapted basis in \(D\) recurrent Lagrange space. (English) Zbl 0787.53020
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 155-170 (1992).
Generalizing the notion of the tangent bundle with a nonlinear connection \(N\) and a tensor field \(M\) of \((1,1)\)-type, the author defines a differentiable manifold \(E(N,M,\nabla,g)\) supplied with an adapted basis \(\delta_ i\), \(\dot\delta_ i\) and \(\widehat{D}x^ i\), \(\widehat{D}y^ i\), a linear connection and a metric tensor \(g\). Putting \(X = \widehat{D}x^ i\delta_ i + \widehat{D}y^ i\dot\delta_ i\), for any tensor field \(T\) on \(E\) the differentiation \(D\) is defined by \(DT = \nabla_ XT\). The space \(E(N,M,\nabla,g)\) in which \(\nabla\) satisfies \(Dg_{ij} = Kg_{ij}\) and \(D\overline{g}_{ij} = \overline{K}\overline{g}_{ij}\) for 1-forms \(K\) and \(\overline{K}\) is called a recurrent Lagrange space. Recurrent Finsler spaces are investigated as subclasses of recurrent Lagrange spaces.
For the entire collection see [Zbl 0764.00002].
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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