Geodesic mappings between Kahlerian spaces

Abstract

Geodesic mappings from a Kahlerian space Kn onto a Kahlerian space Kn will be investigated in this paper. We present a construction of Kahlerian space Kn which admits non-trivial geodesic mapping onto Kahlerian space Kn.

Complete Einstein Metrics are Geodesically Rigid

“…For dimension ≥ 5, the assumption that the metrics are complete is important: if one of them is not complete, one can construct counterexamples (essentially due to [14,37]). For dimensions 3 and 4, (a natural modification of) Theorem 1 is true also locally: Theorem 2.…”

“…By Theorem 2, the result of Petrov remains true for 4-dimensional metrics of other signatures. As we already mentioned in Section 1.1, the counterexamples independently constructed by Mikes [37] and Formella [14] show, that the result of Petrov fails in higher dimensions (so one indeed needs certain additional assumptions, for example the assumption that the metrics are complete as in Theorem 1, which is a standard assumption in problems motivated by physics. )…”

Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

“…Sobchuk, and others [21][22][23][24][25][26][27][38][39][40][41][42][43][44][45][46][47][48]. There are many works devoted to issues of the theory of geodesic mappings, for example [49][50][51][52][53][54][55][56][57][58].…”