ISSN 1055-1344, Siberian Advances in Mathematics, 2018, Vol. 28, No. 2, pp. 101–114.
Allerton Press, Inc., 2018.
Original Russian Text
E.D. Rodionov and V.V. Slavski˘ı, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 120–138.
Polar Transform of Conformally Flat Metrics
E. D. Rodionov
Altai State University, Barnaul, 656049 Russia
Yugra State University, Khanty-Mansiysk, 628012 Russia
Received October 1, 2016
Abstract—In the theory of convex subsets in a Euclidean space, an important role is played by
Minkowski duality (the polar transform of a convex set, or the Legendre transform of a convex
set). We consider conformally ﬂat Riemannian metrics on the n-dimensional unit sphere and their
embeddings into the isotropic cone of the Lorentz space. For a given class of metrics, we deﬁne and
carry out a detailed study of the Legendre transform.
Keywords: Lobachevski ˘ı geometry, convex set, conformally ﬂat metric.
The theory of conformally ﬂat metrics is closely connected to pseudo-Euclidean geometry, which is
due to the existence of a canonical isometric embedding of a conformally ﬂat metric into the isotropic
cone of a pseudo-Euclidean space. This fact was ﬁrst observed by H. B. Brinkmann in  and most
eﬀectively used by N. H. Kuiper in [4, 5].
A conformally ﬂat metric has the form ds
,wheref(x) is a function deﬁned on a subset D of
the Euclidean space. In the present article, we consider only two variants of deﬁnition of a conformally
(1) the spherical model; D is the n-dimensional sphere in the Euclidean space R
and f (x) is
the restriction to the sphere of a homogeneous function f : R
→ R of order 1;
(2) the planar model; D = R
is the n-dimensional Euclidean space and f(x) is an arbitrary
function on D.
Though these two cases are reducible to one another by the stereographic projection, it is convenient
to consider them separately. For the spherical model, in Theorem 3.2, from a given conformally
ﬂat metric ds
of nonnegative one-dimensional curvature, we give the explicit construction
of a quasiconformal diﬀeomorphism of the n-dimensional sphere in a Euclidean space with a sharp
estimate of the quasiconformality coeﬃcient in terms of the curvature of the metric.
2. THE LEGENDRE POLAR TRANSFORM OF A CONFORMALLY FLAT METRIC
The polar metric in the spherical model
Denote by R thereallineandbyR
, the Euclidean
(n +1)-dimensional arithmetic space; let M
× R be the pseudo-Euclidean space, in which
thescalarsquareofavector w =[x,ζ] ∈ M
is deﬁned by w
scalar square of the vector x ∈ R
. Designate as
[x,ζ] ∈ M
: | x|
the upper half of the isotropic cone in M
. In what follows, if it is clear from the context then we
denote x by x.