To the one who added the tag with darij grinberg: I have asked Darij Grinberg and he said that he was not a master in using these methods. He said that I should ask grobber on where he learned this.

Asking for a reference is not a real question? Wherever I ask this question I get no answers. If some users on AoPS can solve problems like this using involutions, then there must be some books which deal with this subject. Where should I ask this if the users on this site cannot give any reference to it?

If this is not the right place for this question, where could I find the answers to my questions?

I don't know if I have the right tags.

@Giuseppe: Thank you for the reference.

Thank you for your answer. I begin to understand how things work in the asked question. I would be interested in some other applications of differential geometry, similar to this one. Any reference would be helpful.