Geometry of Harmonic Identity Maps

TL;DR

This paper investigates the harmonicity of identity maps between Riemannian manifolds with modified metrics involving a smooth function, introducing new examples and a related symmetric tensor field to understand their properties.

Contribution

It extends the concept of harmonic identity maps to metrics altered by a smooth function and introduces a symmetric tensor field linked to their harmonicity.

Findings

Constructed new examples of identity harmonic maps.

Defined a symmetric tensor field related to harmonicity.

Analyzed conditions for harmonicity of identity maps with modified metrics.

Abstract

An identity map $(M,g)\longrightarrow(M,g)$ is a harmonic from a Riemannian manifold $(M,g)$ onto itself. In this paper, we study the harmonicity of identity maps $(M,g)\longrightarrow(M,g-df\otimes df)$ and $(M,g-df\otimes df)\longrightarrow(M,g)$ where $f$ is a smooth function with gradient norm $<1$ on $(M,g)$. We construct new examples of identity harmonic maps. We define a symmetric tensor field on $M$ whose properties are related to the harmonicity of these identity maps.