Existence of suitable diffeomorphisms for pullback geometry

Establish the existence of a diffeomorphism F: R^d -> M into a symmetric Riemannian manifold M that maps a given data manifold in R^d into a low-dimensional geodesic subspace of M while preserving local isometry in a neighborhood of the data manifold (with respect to the ambient Euclidean metric), so that data analysis under the pullback metric g^F is proper, stable, and efficient.

Background

The paper develops a framework for data-driven pullback Riemannian geometry, where a diffeomorphism from Euclidean space to a symmetric Riemannian manifold is used to induce a metric on the data space. The authors show that for proper, stable, and efficient data analysis, such a diffeomorphism should map the data manifold into a geodesic subspace of the target manifold and act as a local isometry near the data manifold.

Having characterized these best practices, the authors turn to the question of constructing such diffeomorphisms via deep learning. Before presenting a heuristic construction strategy, they explicitly note that a rigorous existence result for diffeomorphisms with the required properties remains open. This underscores a foundational gap between the desirable geometric criteria and provable existence conditions.