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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].
##### MSC:
 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
##### Keywords:
recurrent Finsler spaces; recurrent Lagrange spaces