# On isometric correspondence of leaves

## TL;DR

This paper proves that for a generic 3D integrable rolling distribution, establishing an isometric correspondence of leaves necessitates the Bäcklund transformation, excluding certain developable cases.

## Contribution

It demonstrates that Bäcklund transformation is essential for isometric correspondence in generic 3D integrable distributions, extending understanding beyond specific developable cases.

## Key findings

- Isometric correspondence requires Bäcklund transformation in generic cases.
- Excludes developable seed and isotropic developable leaves.
- Provides conditions for isometric correspondence in 3D distributions.

## Abstract

We prove that for a generic $3$-dimensional integrable rolling distribution of contact elements (excluding developable seed and isotropic developable leaves) isometric correspondence of leaves of a general nature (independent of the shape of the seed) requires the B\"{a}cklund transformation.