A Fluid Dynamic Formulation of the Isometric Embedding Problem in Differential Geometry

TL;DR

This paper introduces a novel approach to the isometric embedding problem in differential geometry by formulating it through fluid dynamics equations, revealing a surprising connection between gas dynamics and geometric embeddings.

Contribution

It presents a fluid dynamic formulation of the isometric embedding problem, bridging differential geometry and gas dynamics in a new and insightful way.

Findings

Established a link between gas dynamics and isometric embedding.

Formulated the embedding problem using conservation laws.

Provided a new perspective for solving geometric embedding problems.

Abstract

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.