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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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