graph manifold造句

One of the numerous consequences of the Thurston-Perelman geometrization theorem is that graph manifolds are precisely the 3-manifolds whose Gromov norm vanishes.
This leads to a more modern definition : a graph manifold is either a Solv manifold, a manifold having only Seifert pieces in its JSJ decomposition, or connect sums of the previous two categories.
By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group : at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincar?conjecture.
They proved that there exists a \ varepsilon ( n ) and \ delta ( n ) such that if a 3-dimensional manifold M satisfies { \ rm Vol } ( M ) then one of the following is true : ( i ) " M " is a graph manifold or ( ii ) M has diameter less than \ delta ( n ) and has finite fundamental group.
Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the " positive curvature " geometries "'S "'3 and "'S "'2 ?"'R "', while what is left at large times should have a thick-thin decomposition into a " thick " piece with hyperbolic geometry and a " thin " graph manifold.
It's difficult to find graph manifold in a sentence. 用graph manifold造句挺难的