Diffeomorphisms of pseudo-Riemannian manifolds and the values of the curvature tensor on degenerate planes

TL;DR

This paper investigates how diffeomorphisms that preserve curvature on degenerate planes in pseudo-Riemannian manifolds are characterized as conformal or isometric transformations.

Contribution

It extends the concept of curvature-preserving diffeomorphisms to degenerate planes and classifies them as conformal or isometric under specific conditions.

Findings

Diffeomorphisms preserving curvature on weakly degenerate planes are conformal.

Diffeomorphisms preserving curvature on strongly degenerate planes are isometries.

The paper establishes a link between curvature preservation and geometric transformation types.

Abstract

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degenerate planes) or an isometry (when the condition is fulfilled for strongly degenerate 2-planes).