The purpose of this note is to prove the following
Theorem. A Riemannian manifold is locally conformally flat if and only if
- for , the Weyl tensor vanishes;
- for , the Cotton tensor vanishes.
To this purpose, let us briefly recall some definitions
The Weyl tensor. The Weyl tensor can be defined using the following formula
where and denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors. Writing the Weyl tensor in this way means that the Weyl tensor is actually a (0,4) tensor. It can be seen that the Weyl tensor can be rewritten in this form
where the part
is called the Schouten tensor. We have the following result
Proposition. If , then
The Cotton tensor. The (0,3) tensor above is called the Cotton tensor. Apparently, if either the Weyl tensor or the Ricci tensor vanishes, so does the Cotton tensor.
The Weyl and Cotton tensors under the conformal changes of metric. It is well-known that these tensors are invariant under the conformal changes of metric, that is,
under the change for some smooth function (see this note).
The Riemmanian curvature tensor under the conformal changes of metric. We list here the following rule
See this note for further details.
Locally conformally flat manifolds. Roughly speaking, this is to say at each point , there exists a neighborhood of such that the conformal class of contains the flat metric in , that is to say
We are now in a position to prove the theorem.
Proof of Theorem. We first assume that is locally conformally flat, that is, . If , using the formula for we have
since the Riemmanian curvature tensor vanishes. If , we use the formula for , we obtain
since the Ricci tensor vanishes.
Conversely, if the Weyl tensor vanishes, we have
Under the conformal change, for some , we have
Since the mapping given by is injective, it suffices to show that the following equation
is locally solvable. This can be done using the following whose proof is postponed.
Lemma. Provided the Weyl tensor vanishes, equation
is locally solvable if and only if the following integrability condition is satised
That is, if and only if the Cotton tensor vanishes.
Recall that when ; the condition follows from the Weyl tensor vanishes. This concludes the proof.