Special Weingarten surfaces foliated by circles

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution, one of the Riemann minimal examples, or a generalized cone. Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain     

Linear Weingarten Surfaces in ℝ<Superscript>3</Superscript>

 In this paper we study properties of linear Weingarten immersions and graphs related to non-existence problems and behaviour of its curvatures. The main results are obtained giving a harmonic representation of linear Weingarten surfaces and by proving optimal estimates of the height and curvatures that the immersion must satisfy, characterizing the spherical caps as the only ones achieving these bounds. Received January 25, 2001; in revised form April 4, 2002     

Precomplete Clones on Infinite Sets which are Closed under Conjugation

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latter kind are known.     

Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians

In this paper we give a characterization of real hypersurfaces of type B, that is, a tube over a totally real totally geodesic in complex two-plane Grassmannians with the shape operator A satisfying Aφ + φA = kφ, k is non-zero constant, for the structure tensor φ.     

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.The first author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540075), Ministry of Education, Science, Sports and Culture.The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 14540080), Ministry of Education, Science, Sports and Culture.     

Complete Minimal Hypersurfaces in a Sphere

In this paper we investigate complete minimal hypersurfaces with at most two principal curvatures. We prove that if the squared norm S of the second fundamental form satisfies S ≥ n, then S = n and f(Mⁿ) is a minimal Clifford torus.     

The Boothby-Wang Fibration of the Iwasawa Manifold as a Critical Point of the Energy

In this paper, we show that the Boothby-Wang fibration of the Iwasawa manifold is an unstable critical point for the energy of a distribution. The work of the first author is partially supported by TBAG-?G/2.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

SU(3)-Structures on Hypersurfaces of Manifolds With G 2-Structure

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G₂-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G₂-structure with the shape tensor of the hypersurface. Additionally, we show how to compute the intrinsic SU(3)-torsion and the intrinsic G₂-torsion by means of the exterior algebra.     

Sylvester’s Problem for Symmetric Convex Bodies and Related Problems

We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.     

Locally Symmetric Contact Metric Manifolds

We show that a locally symmetric contact metric space is either Sasakian and of constant curvature 1 or locally isometric to the unit tangent sphere bundle (with its standard contact metric structure) of a Euclidean space. The second author is corresponding author     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.     

Totally Umbilic Isometric Immersions and Curves of Order Two

In this paper we study submanifolds by use of extrinsic shapes of some curves having points of proper order 2, and give a condition that they are totally umbilic. This gives an extension of Nomizu-Yano’s result in [7] on a characterization of extrinsic spheres. The second author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 17540072), Ministry of Education, Science, Sports, Culture and Technology.     

Reiter’s Condition <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> and Approximate Identities for Polynomial Hypergroups

Let K be a commutative hypergroup with the Haar measure . In the present paper we investigate whether the maximal ideals in L¹(K,) have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L(K,). Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.The third author was partially supported by KBN (Poland) under grant 5 P03A 034 20 and by Research Training Network Harmonic Analysis and Related Problems Contract HPRN-CT-2001-00273.     

Geodesics on real hypersurfaces of type A 2 in a complex space form

In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A ₂ in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the ambient complex space form. Author’s address: Toshiaki Adachi, Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).