Surfaces with parallel normalized mean curvature vector

A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceE ^m with parallel normalized mean curvature vector must either lies in aE ⁴ or lies in a hypersphere ofE ^m as a minimal surface. Moreover, it is proved that if a Riemann sphere inE ^m has parallel normalized mean curvature vector, then it lies either in aE ³ or in a hypersphere ofE ^m as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given.     

Vector fields with stable shadowable property on closed surfaces

We prove that the set of vector fields satisfying the C1 stable shadowable property on closed surfaces is characterized as the set of Morse-Smale vector fields. Hence, the vector fields satisfying shadowing property on closed surfaces are C1 dense.     

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S ²ⁿ⁺¹ is the Reeb vector field of a natural Sasakian structure on S ²ⁿ⁺¹. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction. Supported by funds of the University of Lecce and M.I.U.R.(PRIN).     

Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes

A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of A_H . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP ² as well as in complex hyperbolic plane CH ². We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP ² and 41 families in CH ². All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP ² or in CH ² is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Decoupled Potential Integral Equation for Time‐Harmonic Electromagnetic Scattering

We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials A ,φ in the Lorenz gauge, we establish boundary conditions on the potentials themselves rather than on the field quantities. This permits the development of a well‐conditioned second‐kind Fredholm integral equation that has no spurious resonances, avoids low‐frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, φ^scat, is determined entirely by the incident scalar potential φ^inc. Likewise, the unknown vector potential defining the scattered field, A ^scat is determined entirely by the incident vector potential A^inc. This decoupled formulation is valid not only in the static limit but for arbitrary ω ≥ 0$. © 2016 Wiley Periodicals, Inc.     

Real hypersurfaces of a complex projective space

In this paper, we classify real hypersurfaces in the complex projective space C P^\fracn+12C P^{\frac{n+1}{2}} whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.     

Polynomial vector fields on the torus

In this paper it is shown that the structurally stable polynomial vector fields on the torusT ², with singularities, are open and dense in the set of such vector fields. Many kinds of distinct dynamical phenomena are also presented by a list of examples including the Cherry flows. The above result works for analytical vector fields onT ² with the same proof.     

Moving poles of meromorphic linear systems on ℙ<Superscript>1</Superscript>(ℂ) in the complex plane

Let E ⁰ be a holomorphic vector bundle over P¹(C) and †⁰ be a meromorphic connection of E ⁰. We introduce the notion of an integrable connection that describes the movement of the poles of †⁰ in the complex plane with integrability preserved. We show the that such a deformation exists under sufficiently weak conditions on the deformation space. We also show that if the vector bundle E0 is trivial, then the solutions of the corresponding nonlinear equations extend meromorphically to the deformation space.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

Weak convergence to the matrix stochastic integral ∫0 B dB′

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form ∫₀¹ W dW, where W(r) is standard Brownian motion. In multiple regressions and vector autoregressions with vector ARIMA processes, the theory involves weak convergence to matrix stochastic integrals of the form ∫₀¹ B dB′, where B(r) is vector Brownian motion with a non-scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to ∫₀¹ B dB′ under quite general conditions. The theory is applied to vector autoregressions with integrated processes.     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

NONLINEAR HOMOTOPY AND PRE-POINCARÉ LEMMAS

Abstract

Most homotopies considered in the literature are linear homotopies of the form h ⁱ (λ) = λx ⁱ + (1—λ)y ⁱ , 0 ≤ λ ≤ 1. Although these prove to be adequate in most instances, they lack direct geometric significance because {h ⁱ (λ) | 0 ≤ λ ≤ 1} are not orbits of a vector field. On the other hand, the nonlinear homotopy g ⁱ (s) = e ^s x ⁱ + (1—e ^s )y ⁱ ,—∞ ≤ s ≤ 0, are orbits of a vector field (i.e., dg ⁱ /ds = g ⁱ —y ⁱ , g ⁱ (0) = x ⁱ ), and thus have direct geometric significance. This suggests that useful results can be obtained by replacing linear homotopy by transport along flows of smooth vector fields. The purpose of this paper is to elaborate on this simple idea. We define prehomotopy operators induced by vector fields on a manifold. These allow us to obtain finite transport relations and pre-Poincaré lemmas that generalize the classical results. They are shown to reproduce the classical results as asymptotic limits and to obtain representations of all solutions of complete systems of exterior differential equations on a star shaped region of a manifold.     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ²≃ℂ. We describe all representations of \mathfrakg\mathfrak{g} via vector fields in J ⁰ℝ²=ℝ³(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.     

On 2-almost contact tachibana manifolds

LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ _r (r=2m+1, 2m+2) and letη ^r =ξ _r ^b be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure. If the 2-formϕ=η ^2m+1∧η ^2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ^⊥={ξ r} and its complementary orthogonal distributionD ^⊤. On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α^# is a skew-symmetric Killing vector field (see [6]). Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω _ϕ are obtained.     

Characterization of functions as radiation patterns in linear elasticity

In this paper a necessary and sufficient condition for a pair of vector functions to be radiation patterns is presented. More precisely, it is proved that two vector functions, the first in the radial direction and the second in the tangential one, are radiation patterns if and only if there are two entire harmonic vector functions whose radial and tangential projections, respectively, are identical with the previous functions on the unit sphere and whose L²-norm over a sphere of radius R is a function of exponential type in the variable R.     

Null 2-type submanifolds of the Euclidean space E 5 with non-parallel mean curvature vector

Let M be a 3-dimensional submanifold of the Euclidean space E⁵ such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector.