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What is the classification of 3-dim positively curved Alexandrov space? And if a 3-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification?

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I guess you are interested in topological classification (?).

Given a 3-dimensional Alexandrov space $M$, you can always find an other Alexandrov space $\bar M$ with isometric involution $J$ such that

  • $M$ is isometric to $\bar M/J$
  • $\bar M$ is topological manifold without boundary.

This is almost an answer to your first question.

If $M$ is compact and simply connected then $M$ has to be homeomorphic to one of the following (I might miss something):

  • $\mathbb S^3$
  • spherical suspension over $\mathbb R\mathrm P^2$
  • $\mathbb D^3$
  • ball in the cone over $\mathbb R\mathrm P^2$.

If noncompact then you get in addition $\mathbb R^3$ and cone over $\mathbb R\mathrm P^2$.

If the space has a totally quasigeodesic surface then cutting along this surface should give you an Alexandrov space with boundary. Since the curvature is positive it should be disc or ball in cone over $\mathbb R\mathrm P^2$. So you original space is described by an isometric involution of the boundary of one of these spaces.

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