Some results on (k, μ)′-almost Kenmotsu manifolds

In this paper, we study the Weyl conformal curvature tensor 𝒲 and the concircular curvature tensor 𝒞 on a (k, μ)′-almost Kenmotsu manifold M²ⁿ⁺¹ of dimension greater than 3. We obtain that if M²ⁿ⁺¹ satisfies either R · 𝒲 = 0 or 𝒞 · 𝒞 = 0, then it is locally isometric to either the hyperbolic space ?²ⁿ⁺¹ (?1) or the Riemannian product ?ⁿ⁺¹(?4) × ?ⁿ.     

A Ricci inequality for hypersurfaces in the sphere

Let Mⁿ be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere In [9], B. Smyth proved that if Mⁿ, n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering of Mⁿ is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of with nonnegative sectional curvature.Received: 11 November 2003     

On the curvature of minimal submanifolds in a sphere

S. T. Yau proved inAmer. J. Math. 97 (1975), p. 95, Theorem 15 that if the sectional curvature of ann-dimensional compact minimal submanifold in the (n + p)-dimensional unit sphere is everywhere greater than (p – 1)/(2p – 1), then this minimal submanifold is totally geodesic. In this note we improve this bound for the casep 2 to (3p – 2)/(6p).     

Non-compact quadratics-manifolds

The aim of the present paper is the classification of the non — compact quadratics-manifolds.     

The gap between monotone and non-monotone circuit complexity is exponential

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastn ^{c losn} . We observe that this lower bound can be improved to exp(cn ^1/6–o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity, of a graph.     

Calibrations and the size of Grassmann faces

Summary In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR ⁿ . Calibrations correspond to faces of the GrassmannianG(m,R ⁿ ) of orientedm-planes inR ⁿ , viewed as a compact submanifold of the exterior algebra ^m R ⁿ . Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands more general than area. One result implies that form-dimensional surfaces inR ⁿ , with 2 m n – 2, for any integrand , there are -minimizing surfaces with interior singularities.     

Degenerate homogeneous affine surfaces in ℝ3

We study degenerate homogeneous affine surfaces in ³. It is proved that such a surface is either an open part of a plane, a cylinder on an ellipse, parabola or hyperbola or of the surface given byxz – 1/2y ²=0.     

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceM ⁿ in affine spaceR ⁿ⁺¹ equipped with an equiaffine transversal field. IfM ⁿ is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onM ⁿ . This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.Research supported by NSF Grant No. DMS-9101961.     

Codimensionm isometric immersions between pseudo-spheres

We describe all isometric immersionsf:S _p ⁿ S _p ^+1/n+m ,n –p 2m, whose first normal space is -parallel in the complement of totally geodesic points while the set of totally geodesic points does not disconnectS _p ⁿ .     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms

We prove that the universal covering spaces of the generic submanifolds of C P _n and of C H _n are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them.     

The Bernstein problem for affine maximal hypersurfaces

In this paper, we prove the validity of the Chern conjecture in affine geometry [18], namely that an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R ², must be a paraboloid. More generally, we shall consider the n-dimensional case, R ⁿ , showing that the corresponding result holds in higher dimensions provided that a uniform, “strict convexity” condition holds. We also extend the notion of “affine maximal” to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension n≥10. Oblatum 16-IV-1999 & 4-XI-1999?Published online: 21 February 2000     

Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H ³.It is shown that analogs of these equivalences exist for surfaces in S _∞ ² ,the bounding ideal sphere of H ³,leading to a notion of constant mean curvature at infinity of H ³.A parametrization of all complete constant mean curvature surfaces at infinity of H ³ is given by holomorphic quadratic differentials on Ĉ,C, and D.     

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

A characterization for isometries and conformal mappings of pseudo-Riemannian manifolds

Generalizing theorems of Myers-Steenrod and of Hawking, we obtain characterizations for isometries and conformal mappings of pseudo-Riemannian spaces (M, g): Define a local distance function on convex normal neighbourhoods by (p, q) =g(exp _p ^–1 q, exp _p ^–1 q). Then every homeomorphismf locally preserving these functions is an isometry. If (M, g) has indefinite signature andf locally preserves distance zero, it is a conformal diffeomorphism.     

Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

Chebyshev approximation of a point set by a straight line

The problem of calculating the best approximating straight line—in the sense of Chebyshev—to a finite set of points inR ⁿ is considered. First-and second-order optimality conditions are derived and analysed. Lipschitz optimization techniques can be used to find a global minimizer.Communicated by Dietrich Braess.     

Paley-Wiener type theorems on harmonic extensions ofH-type groups

Letn=vz be anH-type group, and letna be the harmonic semidirect product ofn withaR. LetNA be the corresponding simply connected Lie group. If dimv=m and dimz=k, denoteQ=m/2+k. We prove that the spherical Fourier transform is a topological isomorphism between thep-Schwartz spacel ^p (N,A),(0<p2), (0<p2), and the space of holomorphic rapidly decreasing functions on the strip {sC:|Re(s)|<Q/2} with =2/p–1.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.     

On Riemannian g-natural Metrics of the Form a.g s + b.g h + c.g v on the Tangent Bundle of a Riemannian Manifold (M, g)

Let (M, g) be a Riemannian manifold and TM its tangent bundle. In [5] we have investigated the family of all Riemannian g-natural metrics G on TM (which depends on 6 arbitrary functions of the norm of a vector u TM). In this paper, we continue this study under some additional geometric properties, and then we restrict ourselves to the subfamily {G=a.g^s + b.g^h + c.g^v, a, b and c are constants satisfying a > 0 and a(a + c) – b² > 0}. It is known that the Sasaki metric g^s is extremely rigid in the following sense: if (TM, g^s) is a space of constant scalar curvature, then (M, g) is flat. Here we prove, among others, that every Riemannian g-natural metric from the subfamily above is as rigid as the Sasaki metric.