  A Gelfand-Cetlin system is a completely integrable system defined on a
partial flag manifold whose image is a rational convex polytope called a
Gelfand-Cetlin polytope. Motivated by the study of Nishinou-Nohara-Ueda on the
Floer theory of Gelfand-Cetlin systems, we provide a detailed description of
topology of Gelfand-Cetlin fibers. In particular, we prove that any fiber over
an interior point of a k-dimensional face of the Gelfand-Cetlin polytope is an
isotropic submanifold and is diffeomorphic to $(S^1)^k \times N$ for some
smooth manifold $N$. We also prove that such $N$'s are exactly the vanishing
cycles shrinking to points in the associated toric variety via the toric
degeneration. We also devise an algorithm of reading off Lagrangian fibers from
the combinatorics of the ladder diagram.
