Locally conformally flat manifolds and Weyl and Cotton tensors, 2

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 satised

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

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

$\displaystyle \nabla _iX_j=\nabla _jX_i$

which implies $dX = 0$ . Thus locally $X$ is the exterior derivative of some function $f$ . Thus $\nabla f$ solves the equation. From the equation, we have

$\displaystyle\frac{\partial }{{\partial {x^i}}}{X_j} - {X_k}\Gamma _{ij}^k = {S_{ij}} + {X_i}{X_j} - \frac{1}{2}|X{|^2}{g_{ij}}$

$\displaystyle\frac{\partial }{{\partial {x^i}}}{X_j} = {\widetilde c_{ij}} = {S_{ij}} + {X_i}{X_j} - \frac{1}{2}|X{|^2}{g_{ij}} + {X_k}\Gamma _{ij}^k.$

Suppose $p\in M$ and that the coordinates $\{x^i\}$ is dened in a neighborhood of $p$ . The Frobenius theorem says a necessary and suficient condition to locally solve

$\displaystyle\frac{\partial }{{\partial {x^i}}}{X_j} = {\widetilde c_{ij}}$

with $X (p) = X_0$ for any $X_0 \in T_pM$ is the following integrability condition arising from

$\displaystyle\frac{{{\partial ^2}}}{{\partial {x^k}\partial {x^i}}}{X_j} = \frac{{{\partial ^2}}}{{\partial {x^i}\partial {x^k}}}{X_j}$

that

$\displaystyle\frac{\partial }{{\partial {x^k}}}{\widetilde c_{ij}} = \frac{\partial }{{\partial {x^i}}}{\widetilde c_{kj}}.$

More invariantly, the integrability condition arises from

$\displaystyle {\nabla _k}{\nabla _i}{X_j} = {\nabla _i}{\nabla _k}{X_j} + \text{Rm}_{jik}^l{X_l}$

and is

$\displaystyle\begin{gathered} {\nabla _k}{c_{ij}} - {\nabla _i}{c_{kj}} = \text{Rm}_{jik}^l{X_l} \hfill \\ \qquad= (S_i^l{g_{jk}} + {S_{jk}}\delta _i^l - S_k^l{g_{ji}} - {S_{ji}}\delta _k^l){X_l} \hfill \\ \qquad= (S_i^l{g_{jk}}{X_l} - S_k^l{g_{ji}}{X_l} + {S_{jk}}{X_i} - {S_{ji}}{X_k}) \hfill \\ \end{gathered}$

where we have used $W_{jik}^l=0$ and

$\displaystyle\begin{gathered} {\text{Rm}}_{jik}^l = {g^{lm}}{\text{R}}{{\text{m}}_{mjik}} = {g^{lm}}({W_{mjik}} + {(S \odot g)_{mjik}}) \hfill \\ \qquad= {g^{lm}}({W_{mjik}} - {S_{mk}}{g_{ij}} - {S_{ij}}{g_{mk}} + {S_{mi}}{g_{jk}} + {S_{jk}}{g_{mi}}) \hfill \\ \qquad= W_{jik}^l - S_k^l{g_{ij}} - {S_{ij}}g_k^l + S_i^l{g_{jk}} + {S_{jk}}g_i^l. \hfill \\ \end{gathered}$

From the denition of $c_{ij}$ , we obtain

$\displaystyle {\nabla _k}{c_{ij}} = {\nabla _k}{S_{ij}} + {X_j}{\nabla _k}{X_i} + {X_i}{\nabla _k}{X_j} - {X^l}{\nabla _k}{X_l}{g_{ij}}.$

Therefore, we have

$\displaystyle\begin{gathered} {\nabla _k}{c_{ij}} - {\nabla _i}{c_{kj}} = ({\nabla _k}{S_{ij}} + {X_j}{\nabla _k}{X_i} + {X_i}{\nabla _k}{X_j} - {X^l}{\nabla _k}{X_l}{g_{ij}}) \hfill \\ \qquad \qquad- ({\nabla _i}{S_{kj}} + {X_j}{\nabla _i}{X_k} + {X_k}{\nabla _i}{X_j} - {X^l}{\nabla _i}{X_l}{g_{kj}}) \hfill \\ \qquad= ({\nabla _k}{S_{ij}} - {\nabla _i}{S_{kj}}) + ({X^l}{\nabla _i}{X_l}{g_{kj}} - {X^l}{\nabla _k}{X_l}{g_{ij}}) + \hfill \\ \qquad \qquad ({X_j}{\nabla _k}{X_i} + {X_i}{\nabla _k}{X_j} - {X_j}{\nabla _i}{X_k} + {X_k}{\nabla _i}{X_j}) \hfill \\ \qquad= S_i^l{g_{jk}}{X_l} - S_k^l{g_{ji}}{X_l} + {S_{jk}}{X_i} - {S_{ji}}{X_k} \hfill \\ \end{gathered}$

which implies

$\displaystyle {C_{ijk}} = {\nabla _k}{S_{ij}} - {\nabla _i}{S_{kj}} = 0.$

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Ngô Quốc Anh

October 8, 2011