A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in ℍ<Superscript>2</Superscript> × ℝ<Subscript>1</Subscript>

There are examples of complete spacelike surfaces in the Lorentzian product ℍ² × ℝ₁ with constant Gaussian curvature K ≤ −1. In this paper, we show that there exists no complete spacelike surface in ℍ² × ℝ₁ with constant Gaussian curvature K > −1.     

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.     

On graphs without crowns with regularμ-subgraphs

Anm-crown is the complete tripartite graphK _{1, 1,m} with parts of order 1, 1,m, and anm-claw is the complete bipartite graphK _1,m with parts of order 1,m, wherem ≥ 3. A vertexa of a graph Γ is calledweakly reduced iff the subgraph {x є Γ ‖a ^⊥ =x ^⊥} consists of one vertex. A graph Γ is calledweakly reduced iff all its vertices are weakly reduced. In the present paper we classify all connected weakly reduced graphs without 3-crowns, all of whose μ-subgraphs are regular graphs of constant nonzero valency. In particular, we generalize the characterization of Grassman and Johnson graphs due to Numata, and the characterization of connected reduced graphs without 3-claws due to Makhnev. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 874–881, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00462.     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.     

Rigidity of Einstein 4-manifolds with positive curvature

An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε₀, ε₀≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S ⁴, RP ⁴ with constant sectional curvature K=1/3, or CP ² with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S ⁴ and CP ² contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S ⁴, RP ⁴, or CP ² but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions. Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000     

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.     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4

In this paper we give a new definition of harmonic curvature functions in terms of B ₂ and we define a new kind of slant helix which we call quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ and we give a new characterization such as: "a: I ì \mathbb R ? E⁴{``\alpha : I \subset {\mathbb R} \rightarrow E^4} is a quaternionic B ₂–slant helix ${\Leftrightarrow H^\prime_2 -KH_{1} = 0"}${\Leftrightarrow H^\prime_2 -KH_{1} = 0"} where H ₂, H ₁ are harmonic curvature functions and K is the principal curvature function of the curve α.     

The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature K _rad. of an end converges to a constant −1 at infinity with the decay order K _rad. + 1 = o(r ⁻¹) and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order K _rad. + 1 = o(r ⁻¹) is sharp, we will construct a manifold with the radial curvature decay K _rad. + 1 = O(r ⁻¹) and with an eigenvalue \frac(n-1)²4+1{\frac{(n-1)^2}{4}+1} embedded in the essential spectrum [ \frac(n-1)²4, ￥){[ \frac{(n-1)^2}{4}, \infty)} of the Laplacian.     

On exact 2-para Sasakian manifolds

Paracontact and para Sasakian manifoldsM carryingr(1<r≤dimM) Reed vector filds ξ _r have been especially studied by A. Bucki [2], [3], [4]. In the present paper, we consider a (2m+2)-dimensional para Sasakian manifoldM(ϕ, ξ _r , η ^r g), whose Reed convectors η ^r =ξ _r ^b are exact 1-forms and the covariant derivatives of ξ _r are given by ∇ξ _r =f _r dp ^⊺, wheredp ^⊤ means the horizontal component of the soldering formdp andf _r∈C^∞M satisfydf _r =cη ^r ,c=constant. It is proved that such a manifold may be viewed as the local Riemannian productM=M ^⊥×M^⊤, where

Special orthogonal splittings of<Emphasis Type="Italic">L</Emphasis><Stack><Subscript>1</Subscript><Superscript>2<Emphasis Type="Italic">k</Emphasis></Superscript></Stack>

We show that for each positive integerk there is ak×k matrixB with ±1 entries such that puttingE to be the span of the rows of thek×2k matrix [√kI _k,B], thenE,E ^⊥ is a Kashin splitting: TheL ₁ ^2k and theL ₂ ^2k are universally equivalent on bothE andE ^⊥. Moreover, the probability that a random ±1 matrix satisfies the above is exponentially close to 1. Supported by the Israel Science Foundation.     

An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

The upper bound of the $${{\rm L}_{\mu}^2}$$ spectrum

Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e ^h (x)dV(x) be the weighted measure and \triangle_m{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric _m ≥ −(m − 1)K, K ≥ 0, then the bottom of the L_m²{{\rm L}_{\mu}^2} spectrum λ₁(M) is bounded by

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

On the integral of mean curvature of a flattened convex body in space forms

Under the assumptions that E _λ ⁿ is an n-dimensional, simply connected Riemannian manifold of constant sectional curvature λ and L _λ ^r is an r-dimensional, totally geodesic submanifold of E _λ ⁿ , the paper investigates the q-th integral of the mean curvature M _q ⁿ of a convex body K ^r in E _λ ⁿ and gives the expression of M _q ⁿ in the terms of M _p ^r , where M _p ^r is the p-th integral of the mean curvature of K ^r > in L _λ ^r . A result of L. A. Santaló [2] holds in particular.     

Dynamics of rational maps: Lyapunov exponents,bifurcations, and capacity

 Let L(f)=∫log∥Df∥dμ _f denote the Lyapunov exponent of a rational map, f:P ¹→P ¹ . In this paper, we show that for any holomorphic family of rational maps {f _λ :λX} of degree d>1, T(f)=dd ^c L(f _λ ) defines a natural, positive (1,1)-current on X supported exactly on the bifurcation locus of the family. The proof is based on the following potential-theoretic formula for the Lyapunov exponent:

Estimating the diameter of the space of plane convex figures with respect to an affine-invariant metric

A convex figure K ⊂ ℝ² is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that for any two convex figures K₁, K₂ ⊂ ℝ² there is an affine transformation T of the plane such that K₁ ⊂ T(K₂) ⊂ 2.7K₁. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66.     

Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures

Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K ^D of a surface in Euclidean 4-space \mathbb E⁴{\mathbb {E}^4} satisfy K + |K ^D | ≤ H ², where H ² is the squared mean curvature. A surface in \mathbb E⁴{\mathbb {E}^4} is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in \mathbb E⁴{\mathbb {E}^4} form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely classify Wintgen ideal surfaces in \mathbb E⁴{\mathbb E^4} satisfying |K| = |K ^D | identically.