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?
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.