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Lie sphere geometry. With applications to submanifolds. (English) Zbl 0752.53003

Universitext. New York etc.: Springer-Verlag. xii, 207 p. with 14 ill. (1992).
The book under review is an excellent monograph about Lie sphere geometry and its recent applications to the study of submanifolds of Euclidean space. The first two chapters: 1. Lie Sphere Geometry, 2. Lie Sphere Transformations, contain a concise, but comprehensive introduction to the elementary background of the subject. The geometries of hyperspheres in the Euclidean, spherical and hyperbolic spaces are considered a subgeometries of Lie sphere geometry; Moebius and Laguerre geometries are treated in the same way. If a reader knows basic facts about linear algebra and projective geometry, he can use these chapters as a first introduction to sphere geometry.
Chapter 3: Legendre submanifolds, starts with an explicit definition of the contact structure on the manifold of projective lines on the Lie quadric. The integral submanifolds of the contact distribution of maximal dimension are called Legendre submanifolds. Any submanifold of the spaces of constant curvature, mentioned above, induces a Legendre submanifold in a natural way. This and the invariance of certain exterior curvature properties under Lie transformations is the base for the application of Lie sphere geometry to problems about submanifolds in the spaces of constant curvature.
One of the Lie invariant concepts is that of a curvature sphere, which can be introduced for arbitrary Legendre submanifolds. At each point of a Legendre submanifold \(M^{n-1}\) there are at most \(n-1\) distinct curvature spheres. To any curvature sphere belongs a subspace of principal directions of the tangent space \(TM^{n-1}\), which is the direct sum of these principal spaces. If the dimension of such a principal distribution is constant on an open subset \(U\) of \(M^{n-1}\), then the distribution is integrable on \(U\), and its integral submanifolds are curvature surfaces. More generally, a submanifold \(S\) of \(M^{n-1}\) is called a curvature surface, if its tangent spaces \(T_ xS\) are principal spaces of \(M\). Along any curvature surface of dimension \(m>1\) the corresponding curvature sphere is constant. The Lie invariance of curvature spheres allows to generalize the concept of Dupin hypersurfaces \(f: M^{n-1}\to S^ n\) to Dupin submanifolds: these are Legendre submanifolds with the property that along each curvature surface, the corresponding curvature sphere is constant. Lie invariant curvatures of Legendre submanifolds are introduced. The author reviews the theory of isoparametric hypersurfaces and gives a Lie geometric characterization of the corresponding Legendre submanifolds. He proves that tautness is a Lie invariant property.
Finally, he considers compact proper Dupin submanifolds. By a result of Thorbergsson these must be taut, and the number of distinct principal curvatures on them must be 1, 2, 3, 4 or 6, the same as in the case of isoparametric hypersurfaces. The author describes in detail the constructions of compact proper Dupin hypersurfaces not being isoparametric, due to Pinkall and Thorbergsson, and, with another method, due to Miyaoka and Ozawa. These results have been counterexamples to a widely held conjecture.
The last chapter 4 is devoted to the local study of Dupin submanifolds. The method of moving Lie frames is applied to obtain classifications of the Dupin cyclides (proper Dupin hypersurfaces with three principal curvatures) in \(\mathbb{R}^ 4\). The chapter starts with a proof of Pinkall’s theorem about the existence of proper Dupin hypersurfaces in \(\mathbb{R}^ n\) with \(g\) distinct principal curvatures having respective multiplicities \(v_ 1,\dots,v_ g\), \(v_ 1+\dots +v_ g=n-1\). Pinkall’s classification is compared with the author’s own results. The method of half-invariant differentiation, described by G. Bol in his book about projective differential geometry, is adapted to the theory of Legendre submanifolds and applied to the classification of Dupin hypersurfaces.
The book is written in a very clear and precise style. It contains about a hundred references, many comments of and hints to the topical literature, and can be considered as a milestone in the recent development of a classical geometry, to which the author contributed essential results.
Reviewer: R.Sulanke (Berlin)

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A40 Other special differential geometries
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C40 Global submanifolds
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