Geometry of Harmonic Identity Maps

An identity map (M,g)(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)(M,g-df⊗ df) and (M,g-df⊗ df)(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.