Equidimensional isometric maps

TL;DR

This paper introduces a Baire category theorem-based method for constructing isometric maps between Riemannian manifolds, demonstrating that typical 1-Lipschitz maps are isometric in certain extension and restriction problems.

Contribution

It develops a novel approach using Baire category arguments to construct isometries, advancing the understanding of isometric maps in differential geometry.

Findings

Typical 1-Lipschitz maps are isometric in extension problems

The method applies to constructing isometries between Riemannian manifolds

Provides a new perspective on the generic properties of Lipschitz maps

Abstract

In Gromov's treatise Partial Differential Relations (volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 1986), a continuous map between Riemannian manifolds is called isometric if it preserves the length of rectifiable curves. In this note we develop a method using the Baire category theorem for constructing such isometries. We show that a typical $1$-Lipschitz map is isometric in canonically formulated extension and restriction problems.