ON THE METRICS OF THE RIEMANNIAN MANIFOLDS WHICH ADMIT ISOMETRIC IMBEDDING INTO SPACE OF ANY CONSTANT CURVATURE

In this paper the term“Riemannian manifold”means that the fundamental quadraticdifferential form may be indefinite.     

Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature

In this paper, we show that a generalized Sasakian space form of dimension >3 is either of constant sectional curvature, or a canal hypersurface in Euclidean or Minkowski spaces, or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold, or locally a warped product of a real line and a generalized complex space form, or an \({\alpha}\)-Sasakian space form, or it is of five dimension and admits an \({\alpha}\)-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension >5 is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension >3 is also given in this paper.     

The Semi-global Isometric Imbedding in R3 of Two Dimensional Riemannian Manifolds with Gaussian Curvature Changing Sign Cleanly

An abstract Riemannian metric ds²= Edu² + 2Fdudv + Gdv² is given in (u, v) ∈ [0, 2&Pi] × [-&delta, &delta] where E, F, G are smooth functions of (u, v) and periodic in u with period 2&Pi. Moneover K|_{v=0} = 0. K_r|_{v=0} ≠ 0. when> K is the Gaussian curvature. We imbed it semiglobally as the graph of a smooth surface x = x(u, v ), y = y(u, v), z = z(u, v) of R³ in the neighborhood of v = 0. In this paper we show that, if [K_rΓ²_{11}]_{v=0}, and three compatibility conditions are satisified, then there exists such an isometric imbedding.     

On Isometric Immersion of Three-Dimensional Geometries $$\widetilde{SL}_2$$ , Nil,and Sol into a Four-Dimensional Space of Constant Curvature

We prove the nonexistence of an isometric immersion of the geometries Nil ³ and into a four-dimensional space M _c ⁴ of constant curvature c. We establish that the geometry Sol ³ cannot be immersed into M _c ⁴ for c ≠ −1 and find the analytic immersion of this geometry into the hyperbolic space H ⁴ (−1). __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 421–426, March, 2005.     

On the rational recursive sequencex_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }}

The main objective of this paper is to study the boundedness character, the periodic character and the global stability of the positive solutions of the following difference equation $x_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }},n = 0,1,2.....$ where the coefficientsA, B, C, D, α, β, γ, δ, and the initial conditionsx _-3,x _-2,x _-1,x ₀ are arbitrary positive real numbers.     

Application of Soliton Theory to the Construction of Isometric Immersions of M^{{n_{1} }} {\left( {c_{1} } \right)} \times M^{{n_{2} }} {\left( {c_{2} } \right)} into Constant Curvature Spaces M n (±1)

In this paper, we apply the soliton theory to the case of isometric immersion in differential geometry and obtain a family of isometric immersions from M ^{n 1}(c ₁) ×M ^{n 2}(c ₂) to space forms M ⁿ (c) by introducing 2-parameter loop algebra. Received July 14, 1999, Accepted June 15, 2000     

Harmonic tori in De Sitter spaces S^{2n}_1

We show that all superconformal harmonic immersions from genus one surfaces into de Sitter spaces $S^{2n}_{1}$ with globally defined harmonic sequence are of finite-type and hence result merely from solving a pair of ordinary differential equations. As an application, we prove that all Willmore tori in $S^{3}$ without umbilic points can be constructed in this simple way.     

On the difference equationx_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}}

In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.     

On the Number of Solutions of the Equation a_{1}x_{1}^{m_{1}}+\cdots +a_{n}x_{n}^{m_{n}}=bx_{1}\cdots x_{n} in a Finite Field

We obtain an explicit formula for the number of solutions of a special equation in a finite field under a certain restriction on the exponents. Mathematics Subject Classifications (2000) primary: 11G25, secondary: 11T24.     

李奇曲率平行的黎曼流形到欧氏空间的等距浸入

设ｆ：Ｍｎ→Ｒｎ＋ｐ为具平行李奇曲率的黎曼流形到欧氏空间的等距浸入．对ｐ＝１，本文给出了极小条件下以及平均曲率处处非零条件下该浸入的分类     

On the recursive sequencex_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated.     

A short note on a class of Weingarten hypersurfaces in $$mathbb {R}^{n + 1}$$ R n + 1

Geometriae Dedicata - We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying $$H_r = aH + b$$ , for...     

Note on the spectrum of the Hodge-Laplacian for k-forms on minimal Legendre submanifolds in S^{2n+1}

Given a minimal Legendre immersion L in and we prove that is an eigenvalue of the Hodge-Laplacian acting on k and (k-1)-forms on L. In particular we show that the eigenspaces Eig and Eig are at least of dimension Received: 10 February 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

共形空间${\mathbb Q}^{n+1}_1$中类空超曲面的拼挤问题

通过对第二共形基本形式的模长平方作拼挤,给出Clifford超柱面的一个共形特征.     

$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature

In general, the space of $L^p$ harmonic forms $\mathcal{H}^k(L^p(M))$ and reduced $L^p$ cohomology $H^k(L^p(M))$ might be not isomorphic on a complete Riemannian manifold $M$, except for $p=2$. Nevertheless, one can consider whether $\mathrm{dim}\mathcal{H}^k(L^p(M))<+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))<+\infty$. In order to study such kind of problems, this paper obtains that dimension of space of $L^p$ harmonic forms on a hypersurface in unit sphere with finite total curvature is finite, which is also a generalization of the previous work by Zhu. The next step will be the investigation of dimension of the reduced $L^p$ cohomology on such hypersurfaces.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.