Tight and Taut Submanifolds

Contents......Page 2
Preface......Page 6
Remembering Nicolaas Kuiper......Page 8
1. Banchoff\'s two-piece property. Zero-tightness......Page 14
2. Curvature and Knots......Page 32
3. Smooth Submanifolds......Page 39
4. Tight Smooth Surfaces in Codimensions 3 and Higher......Page 50
5. Tightness of Topsets......Page 55
References......Page 61
Tight Submanifolds, Smooth and Polyhedral......Page 64
Introduction......Page 65
1. Tight Surfaces......Page 68
2. Tightness in Higher Dimensions......Page 98
3. Highly Connected Manifolds......Page 111
References......Page 123
1. Introduction......Page 132
2. The Polyhedral Tight Immersion......Page 133
3. Tight Smoothings of Polyhedral Surfaces......Page 135
4. Haab\'s Proof for Smooth Surfaces......Page 141
5. Some Differences Between the Smooth and Polyhedral Theories......Page 143
6. Conclusion......Page 145
References......Page 146
Taut and Dupin Submanifolds......Page 148
1. Taut Submanifolds......Page 150
2. Isoparametric Submanifolds......Page 157
3. Dupin Submanifolds......Page 161
4. Relationship Between the Taut and Dupin Conditions......Page 165
5. Submanifolds in Lie Sphere Geometry......Page 168
6. Compact Proper Dupin Submanifolds......Page 171
7. Local Results on Dupin Submanifolds......Page 175
8. Classi cations of Taut Submanifolds......Page 179
References......Page 184
1. Introduction......Page 194
2. Taut Immersions......Page 200
3. Variationally Complete and Polar Actions......Page 207
4. Equifocal and Weakly Equifocal Submanifolds......Page 211
5. Dupin Submanifolds in a Complete Riemannian Manifold......Page 219
6. Taut Points and Spheres......Page 226
Appendix: Applications of In nite-Dimensional Morse Theory to Section 2......Page 234
References......Page 237
Null-Homotopic Embedded Spheres of Codimension One......Page 242
References......Page 245
Introduction......Page 246
1. Preliminaries......Page 248
2. Hopf Hypersurfaces: When W Is Principal......Page 257
3. Hypersurfaces in Complex Space Forms......Page 267
4. Restrictions on the Shape Operator and the Number of Principal Curvatures......Page 275
5. The eta-Parallel Condition......Page 281
6. Conditions on the Ricci Tensor......Page 284
7. A is W -Parallel......Page 304
8. Additional Topics......Page 308
9. Conclusion and Open Problems......Page 310
References......Page 311
Bibliography on Tight, Taut and Isoparametric Submanifolds......Page 320
Index......Page 350
List of Publications......Page 354
Dissertations of N. H. Kuiper\'s Doctoral Students......Page 362