On leafwise conformal diffeomorphisms (Q1048995)

The paper presents some interesting results concerning conformal diffeomorphisms between 3-dimensional Riemannian manifolds. For every diffeomorphism \(\varphi\) between 3-dimensional Riemannian manifolds, there are two 2-dimensional smooth distribution \(D_+\) and \(D_-\) such that \(\varphi\) is conformal on both of them. The authors give a beautiful proof of the theorem (Theorem 4 in Section 3) about necessary and sufficent conditions for \(\varphi\) to be conformal on a given distribution \(D\). That conditions are equivalent to the necessary and sufficient ones for \(D\) to be one of \(D_+\) or \(D_-\). The paper gives also some integrability conditions of \(D_+\) and \(D_-\). (Theorem 6 in Section 4 and Propositions 7, 8 in Section 5). The last section is devoted to local descriptions of leafwise conformal diffeomorphisms, namely it is shown that it is possible to choose appropriate coordinate systems in which a given leafwise conformal diffeomorphism is holomorphic on the leaves (Theorem 9 in Section 6).