Variation of geometrical quantities under infinitesimal deformation of surfaces

Question

This question is about 2-d surfaces embedded in $\mathbb{R}^{3}$.

It's easy to find information on how the metric $g_{\mu\nu}$ 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.

Answer 1

I would advise you to look at the appendix of this paper, where these issues are addressed. The paper also provides references to the more 'standard' results concerning variations of surfaces.