On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan–Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan–Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.     

On the HSL-flow

We introduce a natural geometric fourth order flow associated to Hamiltonian stationary submanifolds in Kähler–Einstein manifolds. Afterwards we study some of its properties and show short-time existence. In case of Hamiltonian stationary submanifolds with bounded second fundamental form evolving in flat space we obtain an existence time estimate (Theorem 3.7).     

Hyperelliptic functions and geodesic equations

A method for solving geodesic equations in Schwarzschild–de Sitter space–times and higher dimensional Schwarzschild space–times is presented. The solutions are derived from Jacobi's inversion problem on a Riemann surface of genus 2 restricted to the set of zeros of the theta function, which is called a theta–divisor. In its final form, the solutions are given in terms of derivatives of Kleinian sigma functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A kinematic formula for integral invariant of degree 4 in real space form

In this paper,kinematic formulae in a real space form are investigated.A kinematic formula for homogeneous polynomial of degree 4 on the second fundamental forms in a real space form is obtained.The formula obtained is a concrete form of the result of Howard and the analogue of the known formula of Chen and Zhou.     

A generalization of Lelieuvre’s formula to equiaffine immersions of general codimension

A non-degenerate equiaffine immersion of codimension one into an equiaffine space is locally expressed in terms of its conormal map and its affine fundamental form. The expression is called the Lelieuvre’s formula. We recently defined the notions of an equiaffine immersion of general codimension and its transversal volume element map. In this paper, we locally express a non-degenerate equiaffine immersion of general codimension into an equiaffine space in terms of its transversal volume element map and its affine fundamental form.     

Translating solitons of the mean curvature flow

We study some basic properties of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps. Finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. These estimates give rigidity theorems for translating solitons in the Euclidean space in higher codimension.     

Conformal hypersurfaces with the same third fundamental form

We classify all hypersurfaces in a Euclidean space which allow conformal deformations, other than the ones obtained through conformal diffeomorphisms of the Euclidean space, preserving the third fundamental form.     

闵可夫斯基空间中子流形的若干性质

本文研究了Ｆｉｎｓｌｅｒ流形中的子流形,特别地,我们给出了闵可夫斯基空间中超球面的一个特征,同时,我们讨论了闵可夫斯基空间中超曲面的第二基本形式。     

Some characterizations of totally umbilical surfaces in three-dimensional warped product spaces

Surfaces with positive definite second fundamental form in a Riemannian, three-dimensional warped product space are considered. A formula expressing the Gaussian curvature with respect to this new metric on the surface in terms of the Gaussian and mean curvature of the first fundamental form is presented. This formula is then used to give some characterizations of compact, totally umbilical surfaces.     

Sasakian空间型的C-全实伪脐子流形

本文讨论了Ｓａｓａｋｉａｎ空间型的Ｃ-全实伪脐子流形，给出关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

关于空间型中的正螺面

将正螺面引入到一般的空间型中,在Beltrami-Klein坐标系下统一加以表示,并一致地详尽考察其度量、法向、第二基本形式及主曲率,推出正螺面是极小曲面.     

Almost locatedness in uniform spaces

A weak form of the constructively important notion of locatedness is lifted from the context of a metric space to that of a uniform space. Certain fundamental results about almost located and totally bounded sets are then proved.     

Characterization of Weyl operator in terms of Mehler–Fock transform

In this paper, we define the windowed-Mehler–Fock transform and introduce the corresponding Weyl transform. Further, we examine the boundedness of windowed-Mehler–Fock transform in Lebesgue space and establish some of its fundamental properties. Also, we give the criteria of boundedness and compactness of Weyl transform in Lebesgue space.     

Some characterizations of totally umbilical surfaces in three-dimensional warped product spaces

Surfaces with positive definite second fundamental form in a Riemannian, three-dimensional warped product space are considered. A formula expressing the Gaussian curvature with respect to this new metric on the surface in terms of the Gaussian and mean curvature of the first fundamental form is presented. This formula is then used to give some characterizations of compact, totally umbilical surfaces. Postdoctoral researcher of the F.W.O. Vlaanderen.     

四元数射影空间的全实伪脐子流形

给出了四元数射影空间的紧致全实伪脐子流形的关于第二基本形式长度的一个Ｐｉｎｃｈｉｎｇ定理．     

de Sitter空间中具平行平均曲率向量的完备类空子流形

本文证明了ｄｅＳｉｔｔｅｒ空间中具平行平均曲率向量的完备类空子流形在Ｈ２〉Ｃ时其第二基本形式模长平方是上有界的，从而推广了Ｕ－ＨａｎｇＫＩ＾（４）及Ｑ．Ｍ．Ｃｈｅｎｇ＾（３）中的结果。     

Self-similar and oscillating solutions of Einstein's equation and other relevant consequences of a stochastic self-similar and fractal Universe via El Naschie's ε(∞) Cantorian space–time

In this paper we make some suggestion regarding the unification of the fundamental forces and the age of the Universe in the context of the a stochastic self-similar and fractal Universe using El Naschie's ε^(∞) Cantorian space–time. We also show how Einstein's equation can admit for the scale factor a(t) a self-similar solution in agreement with our stochastic self-similar, fractal Universe and El Naschie's ε^(∞) Cantorian space–time. In addition, this solution is found to be oscillating one. Thanks to the first quantization it is possible to recast the equations in a Schrödinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world. Again it appears clear that the Universe and the structures inside must have a memory of its quantum origin as conjectured sometime ago.     

Eine lokale Charakterisierung der Veronesemannigfaltigkeiten

We show that the Veronese submanifold of the standard sphere may be characterised by the property that the second fundamental form is-up to a constant factor-an isometric map from the symmetric product of the tangent space onto the normal space.     

Local imbedding of Einstein spaces

Summary A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces. Entrata in Redazione il 23 giugno 1971.     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.