Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper we prove that the biharmonic conjecture is true for $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.     

Diophantine vectors in analytic submanifolds of Euclidean spaces

Let M be an m-dimensional analytic manifold in R~n.In this paper,we prove that almost all vectors in M (in the sense of Lebesgue measure) are Diophantine if there exists one Diophantine vector in M.     

欧氏空间中子流形的曲率与几何性质

研究了欧氏空间En p中具常数量曲率n(n-1)r的n(n>2)维完备连通子流形,得到了En p截面曲率非负且法联络平坦的完备连通子流形的一个分类定理.     

Non-existence of stable currents in submanifolds of the Euclidean spaces

We prove the non-existence theorems of stable integral currents for certain classes of hypersurfaces or higher codimensional submanifolds in the Euclidean spaces.     

On singularities of submanifolds of higher dimensional Euclidean spaces

Summary Generalizations of principle axes are found for surfaces in E⁴. The singularities generalize umbilics. The generic indicies are computed. For these computations the Thom Transversality Theorem as applied by Feldman to geometry is used. Hower we ? reduce the group ? rendering the calculations more tractible. Also we show that a torus or sphere cannot be immersed in E⁴ with everywhere nonzero curvature of the normal bundle. Entrata in Redazione il 19 novembre 1968.     

An extremal class of conformally flat submanifolds in Euclidean spaces

Summary Let <InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>M^n$ be a Riemannian $n$-manifold with $n\ge 4$. Consider the Riemannian invariant $\sigma(2)$ defined by <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sigma(2)=\tau-\frac{(n-1)\min \Ric}{n^2-3n+4}, $$ where $\tau$ is the scalar curvature of $M^n$ and $(\min \Ric)(p)$ is the minimum of the Ricci curvature of $M^n$ at $p$. In an earlier article, B. Y. Chen established the following sharp general inequality: $$ \sigma(2)\le \frac{n^2{(n-2)}^2}{2(n^2-3n+4)}H^2 $$ for arbitrary $n$-dimensional conformally flat submanifolds in a Euclidean space, where $H^2$ denotes the squared mean curvature. The main purpose of this paper is to completely classify the extremal class of conformally flat submanifolds which satisfy the equality case of the above inequality. Our main result states that except open portions of totally geodesic $n$-planes, open portions of spherical hypercylinders and open portion of round hypercones, conformally flat submanifolds satifying the equality case of the inequality are obtained from some loci of $(n-2)$-spheres around some special coordinate-minimal surfaces.     

Biharmonic PNMC submanifolds in spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ . Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in $\mathbb{S}^{n}$ , their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in $\mathbb{S}^{n}$ is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in $\mathbb{S}^{n}$ .     

Biharmonic hypersurfaces in Euclidean space E5

In this paper, we study biharmonic hypersurfaces in E⁵. We prove that every biharmonic hypersurface in Euclidean space E⁵ must be minimal.     

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 complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

Ricci Soliton Biharmonic Hypersurfaces in the Euclidean Space

Ukrainian Mathematical Journal - We investigate biharmonic Ricci soliton hypersurfaces (Mn, g, ??, ?) whose potential field ?? satisfies certain conditions. We...     

Linear-projection diffusion on smooth Euclidean submanifolds

To process massive high-dimensional datasets, we utilize the underlying assumption that data on manifold is approximately linear in sufficiently small patches (or neighborhoods of points) that are sampled with sufficient density from the manifold. Under this assumption, each patch can be represented (up to a small approximation error) by a tangent space of the manifold in its area and the tangential point of this tangent space.We extend previously obtained results (Salhov et al., 2012 [18]) for the finite construction of a linear-projection diffusion (LPD) super-kernel by exploring its properties when it becomes continuous. Specifically, its infinitesimal generator and the stochastic process defined by it are explored. We show that the resulting infinitesimal generator of this super-kernel converges to a natural extension of the original diffusion operator from scalar functions to vector fields. This operator is shown to be locally equivalent to a composition of linear projections between tangent spaces and the vector-Laplacians on them. We define a LPD process by using the LPD super-kernel as a transition operator while extending the process to be continuous. The obtained LPD process is demonstrated on a synthetic manifold.