Variation of geometrical quantities under infinitesimal deformation

[andresB]

This question is about 2-d surfaces embedded inR3It'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.

 

[pat_connell]

The variation of the second fundamental form is given by the formula $$\delta L_{ij} = \frac{\partial \xi^k}{\partial x^i}L_{kj} + \frac{\partial \xi^k}{\partial x^j}L_{ik} + \xi^k \frac{\partial L_{ij}}{\partial x^k}.$$The variation of the Gauss curvature is given by the formula $$\delta K = \xi^i \frac{\partial K}{\partial x^i} + \frac{1}{2}\left( \frac{\partial^2 \xi^k}{\partial x^i \partial x^j} - \frac{\partial^2 \xi^k}{\partial x^j \partial x^i}\right)L_{ik}L_{jk}.$$The variation of the mean curvature is given by the following formula $$\delta H = \xi^i \frac{\partial H}{\partial x^i} + \frac{1}{2}\left(\frac{\partial^2 \xi^k}{\partial x^i \partial x^j} - \frac{\partial^2 \xi^k}{\partial x^j \partial x^i}\right)L_{ik}L_{jk} + \frac{\partial \xi^k}{\partial x^i}L_{ik} + \frac{\partial \xi^k}{\partial x^j}L_{jk}.$$