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On the geodesic mapping of tangent bundles. (Russian) Zbl 0674.53021
Let M and $$\bar M$$ be smooth manifolds of the same dimension, TM and $$T\bar M$$ the corresponding tangent bundles. Smooth diffeomorphisms $$\Phi: TM\to T\bar M$$ preserving the tangent structures are considered. Assume that $$(M,\Gamma)$$ and $$(\bar M,\bar\Gamma)$$ are spaces with affine connections. The form of an affine connection $$\gamma$$ without torsion on TM in order that the geodesics of TM should be projected via the canonical projection $$\pi: TM\to M$$ on the geodesics of M and that $$(TM,\gamma)$$ should admit a geodesic mapping on $$(T\bar M,^ c{\bar \Gamma})$$ [for this notion, see N. S. Sinyukov, Geodesic mappings of Riemannian spaces (Russian) (Moskva: Nauka 1979; Zbl 0637.53020)] preserving the tangent structures, where $$^ c{\bar \Gamma}$$ is the total lift of $${\bar \Gamma}$$, is established. Some consequences are pointed out.
Reviewer: M.Craioveanu
MSC:
 53B05 Linear and affine connections
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