Isoparametric and Dupin Hypersurfaces

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.Comment: This is a contribution to the Special Issue "Elie Cartan and Differential Geometry", published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM

The linkage relations of six factor loci in guinea pigs

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Assessing Causation in Breast Implant Litigation: The Role of Science Panels

In two recent cases, federal judges appointed panels of scientific experts to help assess conflicting scientific testimony regarding causation of systemic injuries by silicone gel breast implants. This article will describe the circumstances that gave rise to the appointments, the procedures followed in making the appointments and reporting to the courts, and the reactions of the participants in the proceedings

Lie Sphere Geometry and Dupin Hypersurfaces

These notes were originally written for a short course held at the Institute of Mathematics and Statistics, University of São Paulo, S.P. Brazil, January 9–20, 2012. The notes are based on the author’s book [17], Lie Sphere Geometry With Applications to Submanifolds, Second Edition, published in 2008, and many passages are taken directly from that book. The notes have been updated from their original version to include some recent developments in the field. A hypersurface Mn−1 in Euclidean space Rn is proper Dupin if the number of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its principal foliation. The main goal of this course is to develop the method for studying proper Dupin hypersurfaces and other submanifolds of Rn within the context of Lie sphere geometry. This method has been particularly effective in obtaining classification theorems of proper Dupin hypersurfaces

A Characterization of Metric Spheres in Hyperbolic Space by Morse Theory

Let Mn be a differentiable manifold of class C¥. By a Morse function f on Mn, we mean a differentiable function f on Mn having only non-degenerate critical points. A well-known topological result of Reeb states that if Mn is compact and there is a Morse function f on Mn having exactly 2 critical points, then Mn is homeomorphic to an n-sphere, Sn (see, for example, [3], p. 25). In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean n-sphere Sn Ì Rn+p in terms of the critical point behavior of a certain class of functions Lp, p Î Rn+p, on Mn. In that case, if p Î Rn+p, x Î Mn, then Lp(x) = (d(x, p))2 where d is the Euclidean distance function. Nomizu and Rodriguez proved that if Mn (n ³ 2) is a connected, complete Riemannian manifold isometrically immersed in Rn+p such that every Morse function of the form Lp, p Î Rn+p , has index 0 or n at any of its critical points, then Mn is embedded as a Euclidean subspace, Rn, or a Euclidean n-sphere, Sn. This result includes the following: if Mn is compact such that every Morse function of the form Lp has exactly 2 critical points, then Mn = Sn. In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold Mn of hyperbolic space, Hn+p, the spaceform of constant sectional curvature —1