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I'm reading this answer:Relation between mean curvature and conformal metric.

I wonder why we need to use a new orthonormal frame in the new metric to calculate the new second fundamental form?I don't think second fundamental form must be calculated in orthonormal frame.

Any help will be thanked.

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    $\begingroup$ I think the point is that with an orthonormal frame the mean curvature is just $H = \sum_i h_{ii}$ (or $H = \frac{1}{n} \sum_i h_{ii}$, depending on the convention). In a general frame $H = \operatorname{tr}_g A = g^{ij}h_{ij}$, with summation over repeated indices. (Again, depending on your convention, you'd have to normalize by $n$ here too.) $\endgroup$
    – Leo Moos
    Commented Oct 13, 2022 at 16:03
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    $\begingroup$ For that particular answer: read the commends and discussion between the OP and Robert Bryant. The problem is that it is very easy to make a mistake (between covariance and contravariance) in these types of calculations, and sometimes it is easier to re-base using a new frame then trying to remember exactly where the $e^{2f}$ go in every single step. $\endgroup$
    – Willie Wong
    Commented Oct 13, 2022 at 17:59

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