Suppose you have a Riemannian manifold $(M, g)$ and a diffeomorphism $f: M \longrightarrow M$. Define $E^f:= f^*g - g$. I can understand that $E^f$ defines a $(0, 2)$-tensor which in a sense keeps record of everywhere that $f$ fails from being an isometry and by how much it does. However, I came across this other tensor: $$ E^f_C:= \text{tr}(f^*g)g $$
which is called the conformal tensor. I was wondering if you could help with building the intuition of what it is really trying to describe.
You can find the above on pages 16 and 17 of this article: Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds