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 , thenwhere

**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 havesince 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, equationis 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.

what is the relationship between cotton tensor and Bak tensor?

Comment by mathsnail — October 6, 2011 @ 22:21

Sorry, I don’t know. I haven’t heard any Bak tensor.

Comment by Ngô Quốc Anh — October 7, 2011 @ 1:16

I means the Bach tensor, in some reference, both of them are same, but in wiki, Bach tensor is defined by

do you know the difference?

Comment by mathsnail — October 8, 2011 @ 16:08

Comment by mathsnail — October 8, 2011 @ 16:11

The Bach tensor is a tensor of rank 2 while the Cotton tensor has rank 3, so they are different. While the Cotton tensor plays an important role in differential geometry, I just guess the Bach tensor is just a conformally invariant tensor.

Comment by Ngô Quốc Anh — October 9, 2011 @ 9:27