Rigidity of Clifford Minimal Hypersurfaces

In this paper, we prove some rigidity theorems for Clifford minimal hypersurfaces in a unit sphere.Received March 18, 2002; in revised form December 25, 2002 Published online October 15, 2003     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3. The second author was supported by Korea Research Foundation. KRF-2001-015-DP0034, Korea. Received April 26, 2001; in revised form December 17, 2001     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

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.     

Rigidity Theorem for Hypersurfaces in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with H _k = 0 and with two distinct principal curvatures, we give a characterization of torus the . We extend recent results of Perdomo [9], Wang [10] and Otsuki [8].     

C-Totally real pseudo-parallel submanifolds of Sasakian space forms

Let be (2n + 1)-dimensional Sasakian space form of constant ϕ-sectional curvature (c) and M ⁿ be an ⁿ -dimensional C-totally real, minimal submanifold of . We prove that if M ⁿ is pseudo-parallel and , then M ⁿ is totally geodesic.     

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × , where . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

A Practical Criterion For Some Submanifolds To Be Totally Geodesic

Our main theorem is a characterization of a totally geodesic K?hler immersion of a complex n-dimensional K?hler manifold M _n into an arbitrary complex (n + p)-dimensional K?hler manifold by observing the extrinsic shape of K?hler Frenet curves on the submanifold M _n . Those curves are closely related to the complex structure of M _n .     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. (Received 30 December 2000; in revised form 11 April 2001)     

Orlicz-Pettis Polynomials on Banach Spaces

 We introduce the class of Orlicz-Pettis polynomials between Banach spaces, defined by their action on weakly unconditionally Cauchy series. We give a number of equivalent definitions, examples and counterexamples which highlight the differences between these polynomials and the corresponding linear operators. (Received 17 May 1999; in revised form 6 October 1999)     

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.     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

Riemannian submersions with minimal fibers and affine hypersurfaces

Let π : M → B be a Riemannian submersion with minimal fibers. In this article we prove the following results: (1) If M is positively curved, then the horizontal distribution of the submersion is a non-totally geodesic distribution; (2) if M is non-negatively (respectively, negatively) curved, then the fibers of the submersion have non-positive (respectively, negative) scalar curvature; and (3) if M can be realized either as an elliptic proper centroaffine hypersphere or as an improper hypersphere in some affine space, then the horizontal distribution is non-totally geodesic. Several applications are also presented.     

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.     

Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below

Let M be a complete open n-manifold with a base point p, at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface. We discuss the maximal diameter theorem for the compactification of M by attaching the ideal boundary. Under certain conditions we prove that p becomes a pole and that M is isometric to the n-model. Received: 24 September 2000; in final form: 21 November 2001 / Published online: 17 June 2002 Dedicated to Professor Su Bu-Chin on the occasion of his one hundredth birthday The work of the first author was partially supported by the Grant-in-Aid for Scientific Research, No. 12440021 and for Exploratory Research, No. 13874012     

Hypersurfaces of prescribed Gauß curvature in exterior domains

We prove an existence theorem for convex hypersurfaces of prescribed Gau? curvature in the complement of a compact set in Euclidean space which are close to a cone. Received: 23 February 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

<Emphasis Type="Italic">CR</Emphasis>-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N_T ×_fN of a complex submanifold N_T and a totally real submanifold N. There exist many CR-warped products N_T ×_fN in CP^h+p, h = dim_CN_T and p = dim_RN (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N_T is compact. For such CR-wraped products in CP^m (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then N_T is CP^h(4). We also obtain two geometric inequalities for CR-warped products in CP^m with compact N_T.     

Submanifolds of Weyl Flat Manifolds

 We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces. In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds. Received July 16, 2001; in revised form February 6, 2002 Published online August 9, 2002     

A method for the resolution of the Jacobi equation <Emphasis Type="Italic">Y</Emphasis>″ + <Emphasis Type="Italic">RY</Emphasis> = 0 on the manifold <Emphasis Type="Italic">Sp</Emphasis>(2)/<Emphasis Type="Italic">SU</Emphasis>(2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls. Work partially supported by DGI (Spain) and FEDER Projects MTM 2004-06015-C02-01 and MTM 2007-65852 (first author) and by Research Project PGIDIT05PXIB16601PR (second author). Authors’ addresses: A. M. Naveira, Departamento de Geometría y Topología. Facultad de Matemáticas, Avda. Andrés Estellés, N1, 46100 – Burjassot, Valencia, Spain; A. D. Tarrío Tobar, E. U. Arquitectura Técnica, Campus A Zapateira. Universidad de A Coru?a, 15192 – A Coru?a, Spain     

Remarks on η-Einstein unit tangent bundles

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is the space of constant sectional curvature 1 or 2. Authors’ addresses: Y. D. Chai, S. H. Chun, J. H. Park, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea; K. Sekigawa, Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan