Back to harmonic mappings of compact Riemannian manifolds

TL;DR

This paper explores harmonic maps between Riemannian manifolds, providing criteria for harmonicity, analyzing metric decompositions, and studying harmonic forms and metrics to deepen understanding in geometric analysis.

Contribution

It introduces a new criterion for harmonicity of smooth maps, analyzes the structure of pullback metrics, and links harmonic forms and metrics to identity maps in Riemannian geometry.

Findings

Criterion for harmonicity of submersions and diffeomorphisms

Analysis of L2-orthogonal decomposition of pullback metrics

New results on harmonic symmetric bilinear forms and metrics

Abstract

In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing when a smooth submersion or diffeomorphism between Riemannian manifolds is harmonic. This result provides a useful analytic condition for verifying the harmonicity of geometric mappings. Second, we investigate the L2-orthogonal decomposition of the pullback metric associated with a harmonic map. We analyze the structure of this decomposition and discuss its geometric implications, particularly in the context of the energy density and trace conditions. Finally, we study harmonic symmetric bilinear forms and harmonic Riemannian metrics. Special attention is given to their role in the theory of harmonic identity maps. We derive new results that link these notions and demonstrate how they contribute to the broader understanding of harmonicity in geometric analysis.