Hypersurfaces In Spheres Research Articles

The Nonimbeddability of Real Hypersurfaces in Spheres

The Nonimbeddability of Real Hypersurfaces in Spheres

The nonimbeddability of real hypersurfaces in spheres

It is shown that there exist real analytic real hypersurfaces in C n {{\mathbf {C}}^n} which cannot be locally holomorphically imbedded in any finite dimensional sphere S 2 N − 1 ⊂ C 2 N {S^{2N - 1}} \subset {{\mathbf {C}}^{2N}} .

Dust universes with spatial spherical symmetry and euclidean initial hypersurfaces

The Tolman-Bondi equations of pressure-free spherically symmetric systems of particles in general relativity are applied to the special case of a euclidean initial hypersurface.

Automorphisms of Tube Domains and Spherical Hypersurfaces

0. Introduction. Let Q C R' be an open convex set containing no lines and let D = Q X iR' = {x + iy E Cn, y E R' } be the tube domain over U. The domain D is biholomorphically equivalent to a bounded pseudoconvex domain and Aut(D) = {f: D -DI f a biholomorphic map } is a Lie group. When Q is a bounded smooth strictly convex domain, the group Aut(D) was shown ([5]) to consist of real affine linear transformations. In that same paper the Cauchy-Riemann structure of the boundary AD = aQ + iR' was studied in terms of the affine structure of an.

The scalar curvature of minimal hypersurfaces in spheres

The scalar curvature of minimal hypersurfaces in spheres

An algebraic approach to isoparametric hypersurfaces in spheres, II

An algebraic approach to isoparametric hypersurfaces in spheres, II

An algebraic approach to isoparametric hypersurfaces in spheres, I

An algebraic approach to isoparametric hypersurfaces in spheres, I

Isoparametric triple systems of algebra type

Introduction. In this paper we continue our study of isoparametric triple systems. These triple systems have been introduced in [3] and are studied in [3], [4] and [5]. They are in one-to-one correspondence with isoparametric hypersurfaces in spheres which have four distinct principal curvatures. The classes of isoparametric hypersurfaces which have been considered up to now are the homogeneous ones ([10], [11]), the surfaces of FKM-type ([5], [6]), the surfaces satisfying condition (A) and (B) ([9], [10]) and the surfaces where the multiplicity of one of the principal curvatures is < 2 ([12],

Spherical hypersurfaces in complex manifolds

There has recently been a great deal of progress in the mapping or biholomorphic classication theory of domains with strongly pseudoconvex (s. ~. c.) boundaries in C"+l(n> 1). This is based on the combination of results of Chern-Moser [5] on the local differential geometry of real hypersurfaces in C" + 1, and C. Fefferman's marvelous extension theorem [6], which reduces the biholomorphic equivalence of s. ~k. c. domains to the CR equivalence of their boundaries. It is a simple consequence of this work that such domains have an infinite number of "moduli", i.e., there are smooth families of inequivalent such domains depending on arbitrarily many independent parameters (e.g., [3]). Given further restrictive assumptions on the local nature of the boundary, one expects to recover a finite-dimensional problem. In this paper, we make the most drastic assumption possible: we consider domains whose boundaries are everywhere locally CR equivalent to the unit sphere S 2n+l c[~n+l . Such hypersurfaces will be called spherical. The results are of two kinds. On the one hand, we classify the simply-connected spherical hypersurfaces M which are homogeneous under AutcR(M), the automorphism group of the CR structure on M. A consideration of compact space-forms associated to these homogeneous spaces shows the sphere and its quotients by roots of unity are the only compact, homogeneous spherical hypersurfaces. Together with a result of Webster, this completes the classification of compact homogeneous s.~O.c, hypersurfaces in [14]. On the other hand, it is not clear that any of the compact spherical hypersurfaces constructed in the course of the above bound domains in ~,+1, and we, at least, wondered whether S 2"+1 was the only example. In w we give a method for constructing large families of inequivalent such domains in ~"+ 1 Finally, we mention that some interesting global behavior of the chains of E. Cartan and Chern-Moser are observed in examples of w 6 and w 8. Rather more interesting and exotic behavior of chains has been found by Fefferman in [-7], in necessarily less symmetric boundaries than those considered here.

On some types of isoparametric hypersurfaces in spheres, II

On some types of isoparametric hypersurfaces in spheres, II

On some types of isoparametric hypersurfaces in spheres, I

On some types of isoparametric hypersurfaces in spheres, I

Rigidity and convexity of hypersurfaces in spheres

Rigidity and convexity of hypersurfaces in spheres