On a Harmonic Property of the Einstein Manifold Curvature

TL;DR

This paper proves that the curvature 2-form of any Einstein manifold is harmonic, based on the harmonicity condition defined via the de Rham-Lichnerowicz Laplacian, revealing a fundamental property of Einstein manifolds.

Contribution

It establishes that the curvature 2-form of Einstein manifolds is harmonic, linking harmonicity conditions to Einstein geometry for the first time.

Findings

Curvature 2-form of Einstein manifolds is harmonic.

Harmonicity condition is formulated via the de Rham-Lichnerowicz Laplacian.

Provides a new characterization of Einstein manifolds.

Abstract

The harmonicity condition of the curvature 2-form of a pseudo- Riemannian manifold is formulated on the basis of annulment of this form by the de Rham-Lichnerowicz Laplacian. The following theorem is proved: The curvature 2-form of any Einstein manifold is harmonic.