Proper Fredholm submanifolds of Hilbert space
type: Item
instance of: scholarly article
title: Proper Fredholm submanifolds of Hilbert space
zbMATH Open document ID: 0674.58004
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58B15
zbMATH DE Number: 4104020
zbMATH Keywords: submanifold of a Hilbert space
zbMATH Keywords: Fredholm structure
zbMATH Keywords: Dynkin diagram
author: Chuu-Lian Terng
reviewed by: Mihai Anastasiei
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442631
OpenAlex ID: W1566596520
Wikidata QID: Q115183091
DOI: 10.4310/JDG/1214442631
description: scientific article; zbMATH DE number 4104020