Is it in the correct direction, and did anyone do that work? If I differentiate curved surfaces first to a small area and then use a Dihedral group to do reflection on those areas to achieve total symmetry for any curved surface in any dimension, can we classify the reachability of a total symmetric to all curved surfaces by the dihedral group Dn?
"covering space" seems to be a good tool to study the local topological structure and algebraic structure, but it seems can not approach a uniform classification of symmetry of all the asymmetry manifolds (of course i will study more about that theory and if I made a wrong judgment please let me know). I think my idea is more like classifying the degree of symmetry of any asymmetric manifold (by the different order of D group) then analyzing them as symmetric manifolds and classifying their structure generally. Of course I'm not sure if my idea makes sense or not because it still an idea even not an abstract.
The short answer is No and the reason is that diffeomorphisms are local transformations. Discrete transformations can’t be, in general, globally defined.’
hi Nicolis, Yes, I think you made a correct conclusion of ""covering space" seems to be a good tool to study the local topological structure and algebraic structure, but it seems can not approach a uniform classification of symmetry of all the asymmetry manifolds. " seems like you agree with my judgment. However, my idea is to apply it in general by pure algebra, and my problem is I'm not sure my pure algebra is the right way or not. and not sure if i should try to do it or not. Do you or anyone think i should write it done or not or anyone have more advice for me?@Stam Nicolis
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