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I am trying to compute explicitly in terms of extrinsic curvature the self dual part $W^+$ and the anti-self dual part $W^-$ of the Weyl tensor $W$ associated with a codimension 1 submanifold into Euclidean space. I have seen some definitions of $W^+$ and $W^-$ in papers but not their computations(e.g https://arxiv.org/pdf/1809.06339.pdf and https://fdocuments.in/reader/full/self-duality-in-four-dimensional-riemannian-geometry. I just wanted to see what these computations look like.

In four dimension, $W_{abcd}= R_{abcd} +\frac{1}{2}(g_{ac}R_{bd}-g_{bc}R_{ad}+g_{bd}R_{ac}-g_{ad}R_{bc}) +\frac{1}{6}R(g_{ac}g_{bd}-g_{ad}g_{bc})$

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    $\begingroup$ If I understand correctly, you want to find a formula for the self-dual and anti-self-dual parts of the Weyl curvature tensor in terms of the second fundamental form of a hypersurface? Do it one step at a time. Start with the formula for the curvature tensor. From there, calculate the formulas for the Ricci curvature and the scalar curvature. Substitute everything into the formula for the Weyl curvature. You can then calculate the SD and ASD parts. It'll be a long and messy computation, but it's a good way to train yourself in tensor computations. $\endgroup$
    – Deane Yang
    Commented Dec 18, 2020 at 3:56
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    $\begingroup$ I second Deane's recommendation, but I suggest that the best way to do this is to first diagonalize the second fundamental form. Then the calculations will not be messy, and you'll have a better idea of how the AD and ASD parts depend on the eigenvalues than if you do the calculation in general orthonormal frames. Also, once you do this, you can get the general result by abstract nonsense using representation theory. $\endgroup$
    – Robert Bryant
    Commented Dec 18, 2020 at 12:35

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Following the comments above, $$W_{abcd}^+=h_{ac}h_{bd} + \frac{1}{2}[4g_{ac}Hh_{bd}+ g_{bc}h_{kd}h_a^k + 4g_{bd}Hh_{ac}+g_{ad}h_{kc}h_b^k] + \frac{1}{6}[16g_{ac}g_{bd}H^2 +g_{ad}g_{bc}h^2]$$ and then $$W_{abcd}^-=W_{abcd}-W_{abcd}^+$$.

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