Vanishing of top-degree Pontryagin class products obstructs purely electric or magnetic Weyl curvature on compact 4k-manifolds.
Tu.Differential forms in algebraic topology, volume 82 ofGraduate Texts in Mathematics
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A Pontryagin class obstruction for purely electric and purely magnetic Weyl curvature tensors
Vanishing of top-degree Pontryagin class products obstructs purely electric or magnetic Weyl curvature on compact 4k-manifolds.