ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 304–307. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © V. E. Berezovskii, I. Hinterleitner, N. I. Guseva, J. Mikeˇs, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 303–306.
Conformal Mappings of Riemannian Spaces
onto Ricci Symmetric Spaces
V. E. Berezo v skii
, I. Hinterleitner
Uman National University of Horticulture, Ukraine
Brno University of Technology, Brno, Czechia
Moscow State Pedagogical University, Moscow, Russia
Palacky University, Olomouc, Czechia
Received July 24, 2017
Keywords: conformal mapping, Riemannian space, Ricci symmetric space.
Conformal mappings of n-dimensional Riemannian spaces V
have been considered by many
authors. They ﬁnd applications in general relativity theory (see, e.g., –).
In what follows, we assume the signature of the metrics on the spaces V
under consideration to be
arbitrary; i.e., each V
may be a Riemannian space proper or a pseudo-Riemannian space.
Brinkmann reduced the question of whether a Riemannian space V
, n ≥ 3, can be conformally
mapped onto an Einstein space
to the problem of solvability of a nonlinear system of Cauchy-type
diﬀerential equations with covariant derivatives for n +1 unknown functions . This problem was
considered in detail by Petrov in his monograph .
In  and  (see also , , and ), the basic equations of the mappings in question were
reduced to a linear system of Cauchy-type diﬀerential equations with covariant derivatives, which has
made it possible to estimate the number r of arbitrary parameters determining a general solution of the
problem. In other words, the mobility degree of Riemannian spaces with respect to conformal mappings
onto Einstein spaces was found.
In , the ﬁrst lacuna in the distribution of mobility degrees of Riemannian spaces with respect to
conformal mappings onto Einstein spaces was estimated. As is known , the spaces with maximum
mobility degree r = n +2are precisely the conformally ﬂat spaces.
A tensor characterization of nonconformally ﬂat Riemannian spaces for which r = n − 1 has been
obtained. Thus, a sharp bound for the ﬁrst lacuna in the distribution of mobility degrees of Riemannian
spaces with respect to conformal mappings onto Einstein spaces was found, and the maximally mobile
nonconformally ﬂat Riemannian spaces with given degrees of mobility was described.
In these studies, the geometric objects under consideration were assumed to be of suﬃciently high
degree of smoothness.
In , minimal conditions on the diﬀerentiability of geometric objects under conformal mappings of
Riemannian spaces V
onto Einstein spaces were investigated. A system of equations determining these
mappings was found in the form of a closed linear Cauchy-type system with covariant derivatives under
minimum requirements on the diﬀerentiability of the metrics of the conformally corresponding spaces