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**Tight and taut immersions of manifolds.**
*(English)*
Zbl 0596.53002

This book presents material from three related areas in the global theory of immersions into Euclidean spaces and other spaces of constant curvature: tight immersions, taut immersions and isoparametric hypersurfaces. It provides a rather complete and self-contained survey of the most important results in these areas. Most of the material of previous survey articles on these topics is included in this volume. The presentation is sufficiently detailed to be used as a background for a higher course in differential geometry. The reader should have fundamental knowledge in differential geometry and some experience with elements of homology theory and critical point theory. Each of the notions mentioned above is covered by an extra chapter which is presented as self-contained as possible. The introductions to these chapters also contain a survey on related results which were not described there in more detail. Therefore this book can serve as a standard reference for the theory of tight and taut immersions and the geometry of isoparametric hypersurfaces.

The first part of this book deals with tight immersions. After some preliminaries tight maps of compact topological spaces into Euclidean spaces are introduced by the property that preimages of closed half- spaces are homologically simple. Here Cech-homology is used throughout which agrees on triangulable spaces (i.e. on regular sublevels in the differentiable case) with singular homology. Mainly the cases of differentiable or polyhedral immersions are discussed. After some general considerations relations with Banchoff’s two-piece property and minimal total absolute curvature are exhibited. In the case of surfaces in 3- space several examples are outlined. After introducing Kuiper’s top-set analysis some characterization and non-existence results are shown, presenting a rather complete description of the situation in 3-space. The remaining part of this chapter deals with the Chern-Lashof theorem and the characterization of the Veronese manifolds as substantial tight immersion of maximal codimension \(n(n+1)/2\) [see J. A. Little and W. F. Pohl, Invent. Math. 13, 179-204 (1971; Zbl 0217.191)].

Part two studies taut immersions into Euclidean space, introduced by S. Carter and A. West [Proc. Lond. Math. Soc., III. Ser. 25, 701- 720 (1972; Zbl 0242.53029)]. Here tautness defined in analogy to tightness replacing the closed half-spaces by closed balls and complements of open balls. This is a more restrictive property for maps and implies tightness. Relation to special tight embeddings, Banchoff’s spherical two-piece property and characterizations of round spheres are shown at the beginning. For further investigations properties of focal sets of submanifolds are established and Dupin hypersurfaces are introduced in this context, generalizing the well-known cyclides of Dupin. They provide special examples of taut embeddings. Here also some recent results of U. Pinkall and G. Thorbergsson are represented. Then a complete determination of all taut surfaces in Euclidean space is given. The final part deals with investigations due to Carter and West: a characterization theorem for higher-dimensional taut hypersurfaces and a description of totally focal embeddings.

There are two large classes of taut submanifolds: the standard symmetric R-spaces [see M. Takeuchi and S. Kobayashi, J. Differ. Geom. 2, 203-215 (1968; Zbl 0165.249)] and the isoparametric hypersurfaces of spheres. The latter class is characterized by the property of having constant principal curvatures. The study of isoparametric hypersurfaces in spaces of constant curvature has been initiated by E. Cartan (1938) and has been continued by H. F. Münzner (about 1970) and several other authors. For isoparametric hypersurfaces of spheres H. F. Münzner [Math. Ann. 251, 57-71 (1980; Zbl 0417.53030), 256, 215-232 (1981; Zbl 0438.53050)] established several properties of their principal curvatures. Using methods from algebraic topology he proved that the number g of distinct principal curvatures only can be in the set \(\{\) 1,2,3,4,6\(\}\).

Part three of this book mainly is devoted to an exhibition of the first of Münzner’s work [loc. cit.] without presenting the algebraic part of the proof of his main result. This is complemented by a detailed description of some examples due to E. Cartan and K. Nomizu.

The first part of this book deals with tight immersions. After some preliminaries tight maps of compact topological spaces into Euclidean spaces are introduced by the property that preimages of closed half- spaces are homologically simple. Here Cech-homology is used throughout which agrees on triangulable spaces (i.e. on regular sublevels in the differentiable case) with singular homology. Mainly the cases of differentiable or polyhedral immersions are discussed. After some general considerations relations with Banchoff’s two-piece property and minimal total absolute curvature are exhibited. In the case of surfaces in 3- space several examples are outlined. After introducing Kuiper’s top-set analysis some characterization and non-existence results are shown, presenting a rather complete description of the situation in 3-space. The remaining part of this chapter deals with the Chern-Lashof theorem and the characterization of the Veronese manifolds as substantial tight immersion of maximal codimension \(n(n+1)/2\) [see J. A. Little and W. F. Pohl, Invent. Math. 13, 179-204 (1971; Zbl 0217.191)].

Part two studies taut immersions into Euclidean space, introduced by S. Carter and A. West [Proc. Lond. Math. Soc., III. Ser. 25, 701- 720 (1972; Zbl 0242.53029)]. Here tautness defined in analogy to tightness replacing the closed half-spaces by closed balls and complements of open balls. This is a more restrictive property for maps and implies tightness. Relation to special tight embeddings, Banchoff’s spherical two-piece property and characterizations of round spheres are shown at the beginning. For further investigations properties of focal sets of submanifolds are established and Dupin hypersurfaces are introduced in this context, generalizing the well-known cyclides of Dupin. They provide special examples of taut embeddings. Here also some recent results of U. Pinkall and G. Thorbergsson are represented. Then a complete determination of all taut surfaces in Euclidean space is given. The final part deals with investigations due to Carter and West: a characterization theorem for higher-dimensional taut hypersurfaces and a description of totally focal embeddings.

There are two large classes of taut submanifolds: the standard symmetric R-spaces [see M. Takeuchi and S. Kobayashi, J. Differ. Geom. 2, 203-215 (1968; Zbl 0165.249)] and the isoparametric hypersurfaces of spheres. The latter class is characterized by the property of having constant principal curvatures. The study of isoparametric hypersurfaces in spaces of constant curvature has been initiated by E. Cartan (1938) and has been continued by H. F. Münzner (about 1970) and several other authors. For isoparametric hypersurfaces of spheres H. F. Münzner [Math. Ann. 251, 57-71 (1980; Zbl 0417.53030), 256, 215-232 (1981; Zbl 0438.53050)] established several properties of their principal curvatures. Using methods from algebraic topology he proved that the number g of distinct principal curvatures only can be in the set \(\{\) 1,2,3,4,6\(\}\).

Part three of this book mainly is devoted to an exhibition of the first of Münzner’s work [loc. cit.] without presenting the algebraic part of the proof of his main result. This is complemented by a detailed description of some examples due to E. Cartan and K. Nomizu.

Reviewer: Bernd Wegner

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

57Q35 | Embeddings and immersions in PL-topology |