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## Cheng-Yau operator for an electromagnetic 1-form on S4

Suppose we begin with a 1-form $A$ on $S^4(1/h)$ and then construct a stress-energy tensor out of $F=dA$ thereby obtaining a symmetric 2-tensor $\phi$.  Among the many interesting ideas in Cheng and Yau’s paper:  Yau’spaperonSpecialOperatorLikeLaplacian, one is that for symmetric 2-tensors $\phi_{ij}$ satisfying ‘Codazzi conditions’

$\sum_j \phi_{ijj} = 0$

yield a self-adjoint operator (and then they apply it to the second fundamental form tensor for results regarding constant scalar curvature submanifolds).  Their theory can be used for the case of the stress-energy tensor if we can examine whether the ‘Codazzi equation’

$\phi_{ij,k} = \phi_{ik,j}$

can be established.  Now

$T^{ij} = \frac{1}{\mu_0} [ F^{ik} F^j_k - \frac{1}{4} g^{ij} F_{ab}F^{ab}]$

which has well-known properties of symmetry and zero trace.  I don’t understand yet whether the Codazzi conditions are always satisfied for these or whether they are only satisfied for a special class of $A$.