I Variation of geometrical quantities under infinitesimal deformation

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This question is about 2-d surfaces embedded inR3


It's easy to find information on how the metric tensor changes when $$x_{\mu}\rightarrow x_{\mu}+\varepsilon\xi(x)$$

So, what about the variation of the second fundamental form, the Gauss and the mean curvature? how they change?

I found some works on the topic, but, alas, they are expressed very abstractly, so for now they are beyond my understanding.
 

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