Conformally flat submanifolds with flat normal bundle

TL;DR

This paper proves that conformally flat submanifolds with flat normal bundles are locally holonomic and demonstrates how Ribaucour transformations can generate extensive families of such immersions with compatible conformal metrics.

Contribution

It establishes the holonomic nature of these submanifolds and introduces a method to construct related immersions using Ribaucour transformations.

Findings

Submanifolds are locally holonomic with principal coordinate systems.

Ribaucour transformations generate large families of conformally metric-compatible immersions.

The results connect conformal flatness, flat normal bundles, and transformation techniques.

Abstract

We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformal metrics holonomic with respect to the same coordinate system.