On locally convex hypersurfaces in Hadamard manifolds

Locally convex compact hypersurfaces immersed in a hollow simply connected Riemannian space of nonpositive sectional curvature are considered. They are proved to be convex hypersurfaces homeomorphic to the sphere. A similar result for immersed hypersurfaces with nonpositive definite second quadratic form of rank no smaller than one is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 498–507, April, 2000.     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Some Inequalities Involving Two Simplexes

We establish some inequalities on the inradius and circumradius of two simplexes, and prove a number of applications thereof.     

Addendum to “Complete rotation hypersurfaces with <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> constant in space forms”*

Here we study complete rotation hypersurfaces with constant k-th mean curvature H_k in even and 2 < k < n. We prove the existence of a constant such that there are no such hypersurfaces for . We have only one compact hypersurface of this kind with . For each there is a corresponding family of complete immersed rotation hypersurfaces, each family containing two isoparametric hypersurfaces. For H_k ≥ 0, there is also such a family, now containing only one isoparametric hypersurface. Finally, we prove the existence of compact hypersurfaces with arbitrarily large H_k , neither isometric to a sphere nor to a product of spheres. *Bull. Braz. Math. Soc. 30 (2), 1999, 139–161. **Partially supported by FUNCAP, Brazil. ***Partially supported by CNPq, Brazil and DGAPA-UNAM, México.     

Eigenvalue Comparison Theorems on Finsler Manifolds

Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau’s estimation of the first eigenvalue is also given.     

THEOREMS OF BARTH-LEFSCHETZ TYPE ON KAHLER MANIFOLDS WITH PARTIALLY POSITIVE CURVATURE

The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partially positive bisectional curvature without the assumption of nonnegative bisectional curvature. Some applications of the results to holomorphic mappings are given.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.     

Generalized Hadamard Matrices

The paper studies a generalized Hadamard matrix H = (g _{i j}) of order n with entries gi j from a group G of order n. We assume that H satisfies: (i) For m k, G = {g _{m i} g k i ^-1 i = 1,...., n} (ii) g _1i = g _i1 = 1 for each i; (iii) g _ij ^-1 = g _ji for all i, j. Conditions (i) and (ii) occur whenever G is a(P, L) -transitivity for a projective plane of order n. Condition (iii) holds in the case that H affords a symmetric incidence matrix for the plane. The paper proves that G must be a 2-group and extends previous work to the case that n is a square.     

Complete <Emphasis Type="Italic">k</Emphasis>-Curvature Homogeneous Pseudo-Riemannian Manifolds

Abstract. For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalarWeyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova. Mathematics Subject Classification (2000): 53B20     

局部凸空间中的不动点定理

证明了局部凸空间中非凸集上的上半连续凝集值映射的一个Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ型不动点定理,并推广了一些已知的不动点定理。     

On the Kenmotsu Hypersurfaces of Special Hermitian Manifolds

We consider almost contact metric hypersurfaces of almost Hermitian manifolds of class W₃ (in the Gray–Hervella terminology). We establish a criterion for minimality of such hypersurfaces in the case when the contact metric structure is cosymplectic.     

局部凸拓扑向量空间中的不动点定理

本文在局部凸拓扑向量空间中得到了几个新的不动点定理，推广了［２］中的几个重要结果．     

On the Curvatures of Complete Spacelike Hypersurfaces in de Sitter Space

We establish some estimates for the higher-order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in de Sitter space which is contained in certain unbounded regions of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.     

p-Banach Spaces and p-Totally Convex Spaces. II

We show that the results about the set S : ={ [0, 1] ^{1 / p} x + (1 – )^{1 / p} z ^{1 / p} y + (1 – )^{1 / p} z}, where x, y, z elements of a p-absolutely convex space D and `' is a congruence relation on D are the best possible. Finally, we give an explicit construction of the left adjoint of the comparison functor Ô _p : B an _p T C _p (resp. Ô _{p, fin} : V ec _p A C _p ).     

关于Khler流形中管体积的比较定理

本文证明了Kahler流形中围绕复子流形的管体积的比较定理。     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .