On leafwise conformal diffeomorphisms

For every diffeomorphism φ:M→Nφ:M→N between 3-dimensional Riemannian manifolds MM and NN, there are locally two 2-dimensional distributions D±D± such that φφ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of D±D±. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator induced from φ∗φ∗. We investigate the integrability condition of D+D+ and D−D−. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.