The aim of this note is to recall the following interesting result by X. Xu published in Proc. AMS in 1992, here.

Theorem. Suppose is a compact, oriented Riemannian manifold without boundary of dimension . If and , then is constant. In other words, two pointwise conformal metrics that have the same Ricci tensor must be homothetic.

A proof for this result is quite simple. First, we recall the following conformal change

where . Therefore, if , then we obtain the following fact

However, the term appearing in the preceding identity seems to be bad. To avoid it, the author used the following conformal change

for some positive function , i.e. or . Then we calculate to obtain