On leafwise conformal diffeomorphisms

TL;DR

This paper characterizes leafwise conformal diffeomorphisms between 3D Riemannian manifolds, providing algebraic conditions for associated distributions and exploring their integrability and holomorphic properties in suitable coordinates.

Contribution

It establishes necessary and sufficient algebraic conditions for distributions defining leafwise conformality and analyzes their integrability and coordinate representations.

Findings

Conditions for distributions to be leafwise conformal are given.

Integrability criteria for the distributions are derived.

Coordinate systems can be chosen to make leafwise conformal maps holomorphic.

Abstract

For every diffeomorphism $\varphi:M\to N$ between 3--dimensional Riemannian manifolds $M$ and $N$ there are in general locally two 2--dimensional distributions $D_{\pm}$ such that $\varphi$ 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 $(\varphi_{\ast})^*\varphi_{\ast}$. 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.