Surfaces with extreme value curvature in Alexandrov spaces (Q1908310)

An Alexandrov space \(X\) is a locally compact complete length space with curvature bounded either below or above in the distance comparison sense. Consider \(M_1\) be an exponential image of a plane, \(M_2\) be a ruled surface produced by a parallel line field along a geodesic. Then \(\text{curv}_p (X) = k\) \((k\) the bound of the curvature) for \(p \in M_i\), \(i = 1,2\), if and only if \(M_i\) is totally geodesic in \(X\) and locally isometric to the 2-space form of curvature \(k\).