Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

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.     

Number of Singularities of Stable Maps

Sharp estimates for the Ricci curvature of a submanifold M ⁿ of an arbitrary Riemannian manifold N ^n+p are established. It is shown that the equality in the lower estimate of the Ricci curvature of M ⁿ in a space form N ^n+p (c) is achieved only when M ⁿ is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in N ^n+p (c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.      

Strongly taut finitely generated monoids

A strongly taut monoid is a monoid in which all the powers of any element of the monoid have the same elasticity, that is, the ratio between the maximum and the minimum length of the factorizations of an element remains unchanged under powers. We give a procedure to determine if a finitely generated commutative monoid is strongly taut. The authors are supported by the project MTM2004-01446 and FEDER funds. The authors would like to thank the referee and P. Baginski for their heplful comments and suggestions. Author’s addresses: P. A. García-Sánchez, Departamento de álgebra, Universidad de Granada, E-18071 Granada, Espa?a; D. Llena, Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, 04120 Almería, Espa?a; J. C. Rosales, Departamento de álgebra, Universidad de Granada, E-18071 Granada, Espa?a     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

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.     

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     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

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).     

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.     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Anneaux extrémaux dans les surfaces de Riemann

Let X be a closed hyperbolic surface, and let c be a disjoint union of simple closed geodesics on X. A sharp upper bound for the injectivity radius of c is given in terms of the total length of c and of the Euler-Poincaré characteristic of X.

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Résumé Soit X une surface hyperbolique fermée et soit c une réunion disjointe de géodésiques fermées simples de X. Nous établissons une majoration optimale du rayon d’injectivité de c en fonction de sa longueur totale et de la caractéristique d’Euler-Poincaré de X.

     

Affine umbilical surfaces inR 4

Summary A surface in ℝ⁴ is called affine umbilical if for each vector belonging to the affine normal plane the corresponding shape operator is a multiple of the identity. We will classify affine umbilical definite surfaces which either have constant curvature or which satisfy ∇^⊥ g ^⊥. Furthermore, it will be shown that for an affine umbilical definite surface, the affine mean curvature vector can not have constant non-zero length. The last author is a Senior Research Assistant of the National Fund for Scientific Research (Belgium) This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

Rotational linear Weingarten surfaces of hyperbolic type

A linear Weingarten surface in Euclidean space ℝ³ is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ ℝ. Such a surface is said to be hyperbolic when a ² + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ℝ³ that consists of surfaces with self-intersections whose generating curves are periodic. Partially supported by MEC-FEDER grant no. MTM2007-61775.     

Almost Hermitian 4-manifolds of pointwise constant anti-holomorphic sectional curvature

We show that a 4-dimensional almost Hermitian manifold (M, J, g) is of pointwise constant anti-holomorphic sectional curvature if and only if (M, J, g) is self-dual with J-invariant Ricci tensor and K₁₂₁₂ = 0, where K is the complexification of the Riemannian curvature tensor.     

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.