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.
Benkartab et al. (Mon,) studied this question.