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what are possible fundamental and introductory texts about G-structures ? and where i can find the proof of this proposition: if G(group) acts properly discontinuously on a space X , then G is a discrete subset of the space of all continuous functions X--->X with compact-open topology.the converse is false,in general, but is true if X is a complete Riemannian manifold and G is a group of isometries of X.

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I put some more appropriate tags on the question. –  David Roberts Aug 9 '12 at 7:49
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If you do mean $G$-structures in the usual sense (which you may not do, as the rest of your question seems nearly unrelated), another classic text is the last chapter of Lectures on Differential Geometry by Sternberg. He often says things nicely. –  Paul Reynolds Aug 9 '12 at 13:22
    
helpful.thanks Paul. –  DAVID Aug 10 '12 at 5:46

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Concerning the proof that in Riemannian setting proper discontinuity is equivalent to discreteness, even more is true: If $G$ is a subgroup of group of isometries of a Riemannian manifold $M$ so that (for some $x\in M$) $d(x,g_i(x))$ is bounded for a sequence $g_i\in G$, then the sequence $(g_i)$ contains a convergent subsequence (Arzela-Ascoli theorem). Thus, discreteness of $G$ implies proper discontinuity. For your question about $G$-structures: it depends on what you mean: For $(X,G)$-structures, the best reference I know is Thurston's book "Three-dimensional geometry and topology." if you mean $G$-structures as in "reduction of frame bundle to various subgroups $G\subset GL(n,R)$", then read Kobayashi-Nomizu or Kobayashi's "Transformation groups in differential geometry".

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thanks.and where could i find some stuff about the conditions under which the proper discontinuity and discreteness are equivalent ? for example the statement you have just mentioned? –  DAVID Aug 9 '12 at 12:23
    
@David: Do you mean reference to Arzela-Ascoli theorem? If not, then what I wrote is the entire proof: If $G$ is discrete but not properly discontinuous, you find an infinite sequence $g_i$ as in my answer and then get a contradiction with discreteness since the sequence subconverges. (OK, order to apply Arzela-Ascoli you also need to know that in a complete Riemannian manifold, closed metric balls are compact. This would be in any Riemannian geometry textbook, like Kobayashi-Nomizu or Do Carmo, when they prove Hopf-Rinow theorem.) –  Misha Aug 9 '12 at 12:35
    
thanks Misha.it is very helpful. –  DAVID Aug 10 '12 at 5:46

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