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
Hence, the identity becomes
where we also used the following fact
Hence, we have just proved that
and generally that
To go further, the author used the traceless Ricci tensor
Again, the conformal change rule for is given as follows
Since , it is easy to see that
Therefore, one has
Transforming the above identity in terms of gives
This helps us to simplify the former identity as follows
which is nothing but
Integrating this equation over gives showing that mus be constant.
In general, if we denote where
we then have
which then implies
Thus, we have just shown that
Integrating both sides over gives