The push-out space of immersed spheres

Let f: M ^m → ? ^m+1 be an immersion of an orientable m-dimensional connected smooth manifold M without boundary and assume that ξ is a unit normal field for f. For a real number t the map f _tξ: M ^m → ? ^m+1 is defined as f _tξ(p) = f(p) + tξ(p). It is known that if f _tξ is an immersion, then for each p ∈ M the number of the focal points on the line segment joining f(p) to f _tξ(p) is a constant integer. This constant integer is called the index of the parallel immersion f _tξ and clearly the index lies between 0 and m. In case f: $\mathbb{S}^m \to \mathbb{R}^{m + 1} $ is an immersion, we study the presence of a component of index μ in the push-out space Ω(f). If there exists a component with index μ = m in Ω(f) then f is known to be a strictly convex embedding of $\mathbb{S}^m $ . We reveal the structure of Ω(f) when $f(\mathbb{S}^m )$ is convex and nonconvex. We also show that the presence of a component of index μ in Ω(f) enables us to construct a continuous field of tangent planes of dimension μ on $\mathbb{S}^m $ and so we see that for certain values of μ there does not exist a component of index μ in Ω(f).     

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space. Research partially supported by GNSAGA (INdAM) and MIUR of Italy.     

Convexity of spheres in a manifold without conjugate points

For a non-compact, complete and simply connected manifoldM without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres inM is a radial function, then the geodesic spheres are convex. We also show that ifM is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.     

完备Riemann流形中的测地子流形的焦点

利用沿测地线的N-Jacobi场和指标形式,得到了具非负曲率完备Riemann流形中的测地子流形为无焦点的充要条件。     

On self-intersections of immersed surfaces

A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.

     

On a class of Einstein hypersurfaces immersed in a Riemannian manifold

Let x:M→ be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold and let ρ _i  (i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].?If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced by E and P are stated.?The principal curvatures ρ _i are isoparametric functions and the set (ρ₁,...,ρ _n ) defines an isoparametric system [10].?In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P ^♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b ₂≥1. Received: April 7, 1999; in final form: January 7, 2000?Published online: May 10, 2001     

The Ginzburg-Landau manifold is an attractor

Summary The Ginzburg-Landau modulation equation arises in many domains of science as a (formal) approximate equation describing the evolution of patterns through instabilities and bifurcations. Recently, for a large class of evolution PDE's in one space variable, the validity of the approximation has rigorously been established, in the following sense: Consider initial conditions of which the Fourier-transforms are scaled according to the so-calledclustered mode-distribution. Then the corresponding solutions of the “full” problem and the G-L equation remain close to each other on compact intervals of the intrinsic Ginzburg-Landau time-variable. In this paper the following complementary result is established. Consider small, but arbitrary initial conditions. The Fourier-transforms of the solutions of the “full” problem settle to clustered mode-distribution on time-scales which are rapid as compared to the time-scale of evolution of the Ginzburg-Landau equation.     

The mass of an asymptotically flat manifold

We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass, scalar curvature and harmonic maps are described and the positive mass theorem for n-dimensional spin manifolds is proved.     

The rational LS-category of -trivial fibrations

We provide new upper and lower bounds for the rational LS-category of a rational fibration of simply connected spaces that depend on a measure of the triviality of which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if is -trivial for some and enjoys Poincaré duality, then

     

关于''抽象的极小极大定理''的注记

本文指出林壮鹏2000年发表的一个抽象的极大极小定理一文中主要结果的证明需要修正,然后改进了该文的结果,同时给出了一个简单的证明。     

Lightlike real hypersurfaces of indefinite quaternion Kaehler manifold

In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold. Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated. We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions. Received 31 October 2000; revised 20 June 2001.     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

Halforders and automorphisms of chain structures

The concepts of halfordered set, halfordered permutation set and halfordered chain structure will be presented and their interplay investigated. For this purpose the automorphism groups of these structures will be studied. Received 20 October 1999.     

Staircase two-guard kernels of orthogonal polygons

Let S be a simply connected orthogonal polygon in the plane, and assume that S is two-guardable (but not starshaped) via staircase paths. If K is a component of two-kernel (S), then the set of partners of points in K determines a second component K' of two-kernel (S). Thus the components occur in pairs. Moreover, each component is geodesically convex. The results fail without the requirement that S be simply connected. Received 2 November 1999; revised 28 September 2000.     

A variant of the level set method and applications to image segmentation

In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of level set functions are utilized to identify up to phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If phases should be identified, the level set function must approach predetermined constants. We just need one level set function to represent unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.

     

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L ^p -norm of the Gaussian curvature is sufficiently small. Received: 10 June 1999     

The Prime Spectrum of an MV-Algebra

In this paper we show that the prime ideal space of an MV-algebra is the disjoint union of prime ideal spaces of suitable local MV-algebras. Some special classes of algebras are defined and their spaces are investigated. The space of minimal prime ideals is studied as well. Mathematics Subject Classification : 03B50, 06D99.     

关于矩阵的Sharp序、*序和减序

给出短阵sharp序的一个新的刻画，由此得到(半)正定短阵sharp序与其平方矩阵sharp序之间的关系．我们还讨论正规矩阵的*序与减序之间的关系，推广了关于Hermmite矩阵的相应结果．     

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set. Received September 2000/Revised version June 2001