On leafwise conformal diffeomorphisms

For every diffeomorphism $\phi :M\to N$ between 3–dimensional Riemannian manifolds $M$ and $N$ there are in general locally two 2–dimensional distributions $D_{\pm }$ such that $\phi$ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of $D_{\pm }$. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator $({\phi }_{\ast }{)}^{\ast }{\phi }_{\ast }$. We investigate integrability condition of $D_{+}$ and $D_{-}$. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.