zbMATH — the first resource for mathematics

Geodesic mappings of manifolds with affine connection onto symmetric manifolds. (English) Zbl 1380.53020
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 18th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 3–8, 2016. Sofia: Avangard Prima. Geometry, Integrability and Quantization, 99-104 (2017).
Let \(A_n\) and \(\bar{A}_n\) be \(n\)-dimensional manifolds equipped with affine connections. A geodesic mapping is a map \(A_n \to \bar A_n\) mapping geodesics to geodesics. The authors assume \(\bar A_n\) is a symmetric manifold, i.e., its curvature tensor is absolutely parallel, and find necessary and sufficient conditions for the existence of a geodesic mapping \(A_n \to \bar A_n\). More precisely they show that there exists a (locally defined around a point \(x \in A_n\)) geodesic diffeomorphisms \(A_n \to \bar A_n\) if and only if a certain system of PDEs (in explicit form) admits a solution around \(x\). They also remark that the general solution depends on no more than \(n(n+1)\) essential parameters.
For the entire collection see [Zbl 1359.53005].

53B10 Projective connections
53B05 Linear and affine connections
53C35 Differential geometry of symmetric spaces
Full Text: DOI