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

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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

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