Geodesic
Polar transform of conformally flat metrics
Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 120-138.

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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. 
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E. D. Rodionov; V. V. Slavskiǐ. Polar transform of conformally flat metrics. Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 120-138. http://geodesic.mathdoc.fr/item/MT_2017_20_2_a5/

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