# Ngô Quốc Anh

## October 8, 2011

### Locally conformally flat manifolds and Weyl and Cotton tensors, 2

Filed under: Riemannian geometry — Ngô Quốc Anh @ 3:27

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

Lemma. Provided the Weyl tensor vanishes, equation

$\displaystyle {\nabla _i}{\nabla _j}f - {\nabla _i}f{\nabla _j}f + \frac{1}{2}|\nabla f{|^2}{g_{ij}} = {S_{ij}}$

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

$\displaystyle {\nabla _k}{S_{ij}} = {\nabla _i}{S_{kj}}.$

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

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

$\displaystyle {\nabla _i}{X_j} = {c_{ij}} = {S_{ij}} + {X_i}{X_j} - \frac{1}{2}|X{|^2}{g_{ij}},$

where $c = c (X, g)$ is a symmetric 2-tensor depending only on $X$ and $g$. To see this, by the symmetry of the RHS, we have

$\displaystyle \nabla _iX_j=\nabla _jX_i$

## October 4, 2011

### Locally conformally flat manifolds and Weyl and Cotton tensors

Filed under: Riemannian geometry — Ngô Quốc Anh @ 20:45

The purpose of this note is to prove the following

Theorem. A Riemannian manifold $(M^n, g)$ is locally conformally flat if and only if

• for $n \geqslant 4$, the Weyl tensor vanishes;
• for $n=3$, 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

$\displaystyle W = \text{Rm} - \frac{\text{Scal}}{{2(n - 1)n}}g \odot g - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{\text{Scal}}{n}g} \right) \odot g$

where $n\geqslant 3$ and $\odot$ 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

$\displaystyle W = \text{Rm} - \frac{1}{{n - 2}}\left( {\text{Ric} - \frac{g}{{2(n - 2)}}\text{Scal}} \right) \odot g$

where the part

$\displaystyle S = \frac{1}{{n - 2}}\left( {{\text{Ric}} - \frac{g}{{2(n - 2)}}{\text{Scal}}} \right) \odot g$

is called the Schouten tensor. We have the following result