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I heard a conjecture "3-dim positively curved Alexandrov space is of the form S^3/J.(I cannot make sure my statement is accurate). What is the classification of n-dim positively curved Alexandrov space? And if a n-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification? Maybe it's stupid to ask such a big quesiton,I just want to know what we have known about this.can someone recommend some books or papers on it?

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Did not you ask the same question already? –  Anton Petrunin Aug 30 '12 at 23:09
    
Anton - I think the difference in the questions is between '3' and '$n$'. –  HJRW Aug 31 '12 at 12:08
    
Oh, I did not read it to the end --- sorry. –  Anton Petrunin Aug 31 '12 at 12:32

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up vote 4 down vote accepted

Again, I guess you want a topological classification. Such classification would include classification of all smooth positively curved manifolds which is too much to ask.

For the (quasi)geodesic subset, one should be able to say something if it has big dimension, say codimension 1; otherwise there is no chance.

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I have read your paper"parallel transportation".You had extended Frankel's theorem to Alexandrov space.I am now considering extended Wilking's connectivity theorem to Alex.I have read some of your book"Alexandrov geometry",your paper "semiconcave functions",but topic on positive curved Alex is so limit,can you recommend some papers on it? –  jiangsaiyin Sep 1 '12 at 6:56

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