On the Theory of Geodesic Mappings of Einstein Spaces and their Generalizations

Complete Einstein Metrics are Geodesically Rigid

“…The proof from [21,39] is very complicated: they prolonged (= covariantly differentiated) the basic equations (8) 6 times , and used the condition that the metric is Einstein at every stage of the prolongation.…”

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…This establishes part (i) and part (ii) is immediate from part (i). This result leads naturally to the next theorem which is similar to a result in [6,7]. The proof of this theorem gives a nice example of a situation where, unlike many of those in the previous section, non-trivial solutions of the Sinyukov equation (15) arise and how the inversion of the corresponding pair (a, λ) to get the pair (g ′ , ψ) is carried out.…”

“…There has been some recent interest in the study of projective relatedness of (metric) connections and, in particular, within Einstein's theory [1,2,3,4,5,6,7,8,9]. Thus, roughly speaking, one assumes that one has two Lorentz metrics on a given space-time M whose Levi-Civita connections give rise to the same set of geodesic paths (unparametrised geodesics) on the space-time manifold and then tries to find the relationship between these metrics and connections.…”

On the Equations of Conformally-Projective Harmonic Mappings

“…f is represented by the identity map from U toŪ; see, for example, [2,4,6,8,11,12]. In U and U we introduce a common coordinate system x with respect to the diffeomorphism f , so that the point M ∈ U and its image f (M) ∈Ū have the same coordinates x = (x 1 , x 2 , .…”