Proper Fredholm submanifolds of Hilbert space
Let M be a submanifold of a Hilbert space V, \(\nu\) (M) be its normal bundle and Y be the map from \(\nu\) (M) to V defined by \(Y(v)=x+v\) if \(v\in \nu (M)_ x\). The submanifold M is called proper Fredholm (PF) if the restriction of Y to the unit disk normal bundle is proper. The author develops a rich and interesting theory of PF submanifolds. He proves that every PF submanifold has a natural Fredholm structure and the shape operators are compact. Then he studies the geometry of PF submanifolds with flat normal bundle, especially the focal structure, the curvature distributions, curvature normals and gives a useful example from infinite dimensional Lie theory. He associates to each isoparametric PF submanifold M a marked affine Dynkin diagram and shows that the homology of M can be computed explicitly from its Dynkin diagram, proving that every nondegenerate Euclidean distance function \(f_ a\) on M is a perfect Morse function, that M is taut and that the set of critical points of \(f_ a\) is a W-orbit.
scientific article; zbMATH DE number 4104020