Differential Topology Interacts with Isoparametric Foliations

Problems related to isoparametric theory

“…We call such a manifold N to have a DDBD structure (D(E ± ), ϕ), which is then equivalent to saying that N has an isoparamtric foliation when N is a closed simply connected manifold (cf. [35]).…”

“…Although we have arrived at a celebrated ending of the long way of classification for isoparametric hypersurfaces in unit spheres since from E. Cartan in the late 1930s', we find that we are standing at a new beginning of exploring applications of this theory as well as an even longer way of classification for isoparametric hypersurfaces in general Riemannian manifolds such as homotopy spheres. For details we refer to the excellent book [4] and the surveys [8,35,38,50].…”

“…Two isoparametric foliations (N 1 , F 1 ) and (N 2 , F 2 ) are called equivalent if there is a foliated diffeomorphism between them, i.e., there is a diffeomorphism from N 1 to N 2 which maps each leaf of F 1 to that of F 2 . For details we refer to the survey [35]. Applications to other areas such as 4 dimensional smooth Poincaré conjecture, 4-manifolds geometry, closed geodesic and exotic smooth structures, etc., have been achieved by [17,15,51,14].…”