Differentiable Manifold Of Dimension Research Articles

Notes on some properties of the natural Riemann extension

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

The Sasaki-Kenmotsu manifolds

In the present paper, we introduce a new class of structures on an even dimensional differentiable Riemannian manifold which combines, well known in literature, the Sasakian and Kenmotsu structures simultaneously. The structure will be called a Sasaki–Kenmotsu structure by us. Firstly, we discuss the normality of the Sasaki–Kenmotsu structure and give some basic properties. Secondly, we present some important results concerning with the curvatures of the Sasaki–Kenmotsu manifold. Finally, we show the existence of the Sasaki–Kenmotsu structure by giving some concrete examples.

Semi-Simplicity of a Lie Algebra of Isometries

A spray S on the tangent bundle TM with a n dimensional differentiable manifold M defines an almost product structure Γ (Γ2 =I, I being the identity vector 1-form) and decomposes the TTM space into a direct sum of horizontal space (corresponding to the eigenvalue +1) and vertical space (for the eigenvalue -1). The Lie algebra of projectable vector fields whose Lie derivative vanishes the spray S is of dimension at most n2 +n. The elements of the algebra belonging to the horizontal nullity space of Nijenhuis tensor of Γ form a commutative ideal of . They are not the only ones for any spray S. If S is the canonical spray of a Riemannian manifold, the symplectic scalar 2-form Ω which is the generator of the spray S defines a Riemannian metric g upon the bundle vertical space of TM. The Lie algebra of infinitesimal isometries which is written contained in is of dimension at most . The commutative ideal of is also that of . The Lie algebra of dimension superior or equal to three is semi-simple if and only if the nullity horizontal space of the Γ Nijenhuis tensor is reduced to zero. In this case, Ag is identical to . Mathematics Subject Classification (2010) 53XX • 17B66 • 53C08 • 53B05ns.

Properties of Modified Riemannian Extensions

Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $% \widetilde{g}_{\nabla,c}$ on $T^{\ast}M$ defined by means of a symmetric $% (0,2)$-tensor field $c$ on $M.$ We get the conditions under which $T^{\ast}M $ endowed with the horizontal lift $^{H}J$ of an almost complex structure $J$ and with the metric $\widetilde{g}_{\nabla,c}$ is a K\{a}hler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric $\widetilde{g}_{\nabla,c}$ are presented.

Vector optimal control problem on a differentiable manifold: A realistic approach

The present paper discusses vector optimal control of a functional whose domain of definition is a product manifold M × N, where M is a differentiable manifold of dimension M and N is a differentiable manifold of dimension N with differentiable variety as its boundary respectively; N can be viewed as a set of control parameters. It also gives a new concept viz. vector valued semi optimal control of a functional on the said restricted domain. Finally, the paper establishes some examples in favour of such type of optimal control problem.

ALMOST PSEUDO CONTACT STRUCTURE

A new kind of structure is introduced in an even dimensional differentiable Riemannian manifold and some basic properties of this structure is discussed. Also the existence of such structure is shown with an example.

MYLLER CONFIGURATIONS IN FINSLER SPACES. APPLICATIONS TO THE STUDY OF SUBSPACES AND OF TORSE FORMING VECTOR FIELDS

In this paper we define a Myller configuration in a Finsler space and use some special configurations to obtain results about Finsler subspaces. Let F n =(M, F ) be a Finsler space, with M a real, differentiable manifold of dimension n. Using the pull back bundle (π∗TM, π, TM) of the tangent bundle (TM, π, M) by the mapping π = π/TM and the Cartan Finsler connection of a Finsler space, we obtain an orthonormal frame of sections of π∗TM along a regular curve in TM and a system of invariants, geometrically associated to the Myller configuration. The fundamental equations are written in a very simple form and we prove a fundamental theorem. Important lines in a Finsler subspace are defined like special lines in a Myller configuration, geometrically associated to the subspace: auto parallels, lines of curvature, asymptotes. Torse forming vector fields with respect to the Cartan Finsler connection are characterized by means of the invariants of the Frenet frame of a versor field along a curve, and the new notion of torse forming vector fields in the sense of Myller is introduced. The particular cases of concurrence and parallelism in the sense of Myller are completely studied, for vector fields from the distribution T m of the Myller configuration and also from the normal distribution T p.

On the existence of paratactical three-webs

We prove the existence of paratactical 3-webs W(3, 2, r) of codimension r ≥ 2 on a (2r)-dimensional differentiable manifold by constructing a series of examples of such webs.

The Weyl bundle as a differentiable manifold

Construction of an infinite dimensional differentiable manifold ${\mathbb R}^{\infty}$ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra and a Weyl algebra bundle are presented. Continuity of the $\circ$-product in the Tichonov topology is proved. Construction of the $*$-product of the Fedosov type in terms of theory of connection in a fibre bundle is explained.

Generalized Homotopy Approach to Multiobjective Optimization

This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.

The Space of Parabolic Affine Sphereswith Fixed Compact Boundary

The aim of this paper is to study the moduli space of solutions of the Dirichlet problem associated to the equation of Monge-Ampeere type, det , on an exterior planar domain. We prove that this moduli space is either empty or a differentiable manifold of dimension five.

Quantum integrability for the Kovalevsky top

LetN be a differentiable manifold of dimension n. LetM = T ∗N be the cotangent bundle of N , and denote by σ the natural symplectic form on M . In local coordinates q = (q1, . . . qn) on N and pi = ∂ ∂qi (viewed as function on M) we have σ = ∑ dpi∧dqi. For f ∈ C ∞(M) the hamiltonian vector field vf on M is defined by df(·) = −σ(vf , ·). For this sign convention we have [vf , vg] = v{f,g} (1.1) with the Poisson bracket {f, g} of f, g ∈ C∞(M) defined by {f, g} = vf (g) = σ(vf , vg), or in local coordinates {f, g} = ∑ (

A theorem on polynomial lorentz structures

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).We shall suppose that for any x ∈ Mis the minimal polynomial of the endomorphism fx: TxM → TxM.We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signaturesuch that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

A continuation principle for forced oscillations on differentiable manifolds

In the present paper we are concerned with the existence of Γ-periodic solutions for the differential equation x(t) = f(t, .x( r)), t e R, where / is a continuous time dependent Γ-periodic tangent vector field defined on an ^-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (l/T)fίff(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation x(t) = λf(ί,x(t)), /eR and λ £ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a Γ-periodic solution. 0. Introduction. Let M be an w-dimensional differentiable manifold, possibly with boundary, and let /: RxM->Γ(M) be a time dependent Γ-periodic tangent vector field on M. In this paper we give a topological result concerning the existence of Γ-periodic solutions for the differential equation

The Kervaire invariant of immersions

where f2. G is an appropriate cobordism theory of immersed submanifolds of Euclidean space. By generalizing techniques of Browder [2] we shall give necessary and sufficient Adams spectral sequence conditions for an element in t2. G to have nonzero Kervaire invariant. Also we will prove that for every j > 1 there exists a closed, differentiable manifold of dimension 2 ) § together with an immersion in R2s§ which has nonzero Kervaire invariant. Before we state our results more precisely, we first recall some preliminaries about the Kervaire invariant. The classical Kervaire invariant of stably framed corbodism, K: f24kf + 2---'Z/2, is the obstruction to a framed cobordism class containing a framed homotopy sphere. The question of the existence of smooth, stably framed, closed manifolds with nonzero Kervaire invariant has intrigued topologists since the early 1960's. By using the Thom-Pontr jagin construction which equates framed cobordism, f2 fr, with the stable homotopy groups of spheres, n,(S~ W. Browder gave necessary and sufficient conditions on the mod 2 Adams spectral sequence f r s 0 for there to exist elements in f 2 . r c . ( S ) which have Kervaire invariant one. In [2] he proved the following.

Differential forms and nuclear structure

The solution of a set of Schrodinger’s equation can be considered from the point of view of integration of differential forms over simplices in higher dimensional differentiable manifolds. In this way, the problems of nuclear structure are transferred from conjecturing potential wells and nuclear models to one of integration using the appropriate simplices. Using this approach, it is shown that without any reference to physical models it is possible to obtain the neutron-proton ratio of stable spinless nuclei their coulomb and binding energies, which is in good agreement with that obtained by experiment. It is also shown from the nature of the solution that a nucleus consists of only two types of particles, one having an energy of the coulomb type and the other neutral. Possible extension to the theory for nuclear systems having spin is also discussed.

Pontrjagin numbers and pseudo-riemannian geometry

An integral formula for the Pontrjagin numbers of a compact orientable real 4k dimensional differentiable manifold which has a pseudo-Riemannian metric is derived. This formula allows the Pontrjagin numbers to be expressed in terms of the index, or signature, of the differentiable manifold. The application of these formulae to the four dimensional Lorentzian manifolds of the general theory of relativity is discussed. A corresponding formula for the Chern numbers of a complex differentiable manifold with a Hermitian metric is also given.

Homotopy theory of infinite dimensional manifolds

In the past several years there has been considerable interest in the theory of infinite dimensional differentiable manifolds. While most of the developments have quite properly stressed the differentiable structure, it is nevertheless true that the results and techniques are in large part homotopy theoretic in nature. By and large homotopy theoretic results have been brought in on an ad hoc basis in the proper degree of generality appropriate for the application immediately at hand. The result has been a number of overlapping lemmas of greater or lesser generality scattered through the published and unpublished literature. The present paper grew out of the author's belief that it would serve a useful purpose to collect some of these results and prove them in as general a setting as is presently possible.

Contact hypersurfaces in certain Kaehlerian manifolds

Introduction. An odd dimensional differentiable manifold M is called an almost contact manifold if the structure group of its tangent bundle can be reduced to (7(n)xl, where U(n) means the real representation of the unitarygroup of n complex variables. Recently S. Sasaki [12] proved that the condition of almost contact manifold can be represented by a set of tensor fields φ, f, η which satisfy certain conditions. From this fact it is known that any orientable hypersurface of an almost complex manifold admits the structure of an almost contact manifold. The study of hypersurfaces of an almost complex manifold is perhaps one of the most fruitful aspects of the theory of an almost contact manifold. In fact, M. Kurita [8], Y. Tashiro and S. Tachibana [15] and the present author [9,10,11] proved many theorems on certain hypersurface of an almost complex manifold. If the induced almost contact metric of a hypersurface of almost Hermitian manifold is contact metric we call the hypersurface a contact hypersurface. In this paper, we mainly study the differential geometric properties of contact hypersurfaces of Kaehlerian manifold of constant holomorphic sectional curvature. In particular, as an even-dimensional Euclidean space E can be regarded as a Kaehlerian manifold with vanishing sectional curvature, the results obtained apply to contact hypersurface of a Euclidean space. In §1, we give first of all the definition of an almost contact metric manifold and, in §2, a short summary of those parts of the theory of hypersurfaces in an almost Hermitian manifold which are necessary for what follows. Moreover in §2 some general properties of contact hypersurfaces of a Kaehlerian manifold are derived and in §3 some examples of contact hypersurfaces of a Kaehlerian manifold are exhibited. After these preliminaries we consider in §4 hypersurfaces of Kaehlerian manifold of constant holomorphic sectional curvature and prove that there are at most three distinct principal curvatures of a contact hypersurface in

Remarks on $4$-dimensional differentiable manifolds

Remarks on $4$-dimensional differentiable manifolds