But the sign on the inverse is the reason for your confusion. They are either sloppy about the sign, or have an index flipping convention that fixes it up, I don't know. The result is that the entries of the upper index $\omega$ are the inverse matrix (up to a sign) of the lower index $\omega$, and this is what the authors are trying to say. Since the g's are inverse to each other, so you get cancellation in the middle. ![]() ![]() ![]() Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ \omega^l_k = -\delta^i_k $$
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