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For complex manifolds , people usually write the first fundamental form as $ds^2=g_{a\bar{b}}dz^ad\bar{z}^b$ (at least physicists) with a bar over the second index of the metric, but don't usually write the bar over the second one-form $d\bar{z}^b$. I am not sure why this is done. As I see it, for the notation to be consistent one should write $ds^2=g_{a\bar{b}}dz^ad\bar{z}^{\bar{b}}$. Can someone clarify this. Thanks

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    This is an amusing conundrum I had never thought about, even though I like complex manifolds! Perhaps a more logical notation would be $\quad ds^2=g_{a\bar{b}}dz^ad\bar{z^b}$ , where the second index $\bar b$ of $g$ is just the integer $b$ overlined for mnemotechnical reasons and the last symbols means "take the differential of the complex conjugate of the $b$-th complex variable".2011-04-04
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    putting the bar only over the index of the metric is a fast way of showing that you are considering the manifold as a complex manifold, without saying it directly. maybe thats the origin of the notation, but I am not sure.. Btw, nice notation!2011-04-04
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    On a lighter note, tread carefully when using the Einstein summation convention suppressing the $\Sigma$: some mathematicians really hate it and get furious when seeing it used . Although, to tell the truth, I don't use the convention either, it doesn't bother me and I wonder why it can elicit such strong reactions. Maybe one of those sigmaphiles will enlighten us...2011-04-04

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