Locally conformally flat manifolds and Weyl and Cotton tensors, 2

The purpose of this note is to prove the following result that left in the previous entry

Lemma. Provided the Weyl tensor vanishes, equation

is locally solvable if and only if the following integrability condition is satised

That is, if and only if the Cotton tensor vanishes.

Proof. It is necessary and suffcient to find a 1-form locally such that

where is a symmetric 2-tensor depending only on and . To see this, by the symmetry of the RHS, we have

Locally conformally flat manifolds and Weyl and Cotton tensors

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