Flatness produced by some geometric PDEs

TL;DR

This paper explores various geometric PDEs related to flatness conditions in differential geometry, offering new perspectives and methods for analyzing connection, curvature, Ricci, and scalar curvature flatness.

Contribution

It introduces a novel approach to studying geometric PDEs of flatness using Euler-Lagrange prolongations and least squares functionals, providing fresh insights in differential geometry.

Findings

New theorems on flatness conditions

Unified framework for geometric PDEs

Application of least squares Lagrangian densities

Abstract

This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems about flatness introduce a different point of view in Differential Geometry. The second idea is to introduce and study the Euler-Lagrange prolongations of PDEs-flatness solutions via associated least squares Lagrangian densities and functionals on Riemannian manifolds. All geometric PDEs turned into one of the most intensively developing branches of modern differential geometry.