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etotheipi

There's an integral over a 2-sphere ##S## with unit normal ##N^a## within a hypersurface orthogonal to a Killing field ##\xi^a##$$F = \int_S N^b (\xi^a / V) \nabla_a \xi_b dA = \frac{1}{2} \int_S N^{ab} \nabla_a \xi_b dA, \quad N^{ab} := 2V^{-1} \xi^{[a} N^{b]}$$which follows because the Killing equation is ##\nabla_{a} \xi_b = \nabla_{[a} \xi_{b]}## and we can also write ##\xi^a N^b \nabla_{[a} \xi_{b]} = \xi^a N^b \delta^{[c}_{a} \delta^{d]}_b \nabla_c \xi_d = \xi^{[c} N^{d]} \nabla_c \xi_d##. The original integral is supposed to transform into$$F = \frac{-1}{2} \int_S \epsilon_{abcd} \nabla^c \xi^d$$but I don't see how yet. Can anyone provide a hint? Thanks.

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