Odd-dimensional Euclidean Space Research Articles

The Willmore energy and the magnitude of Euclidean domains

We study the geometric significance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domainXXin an odd-dimensional Euclidean space, we show that the asymptotic expansion of the functionMX(R)=Mag(R⋅X)\mathcal {M}_X(R) = \mathrm {Mag}(R\cdot X)atR=∞R = \inftydetermines the Willmore energy of the boundary∂X\partial X. This disproves the Leinster-Willerton conjecture for a compact convex body in odd dimensions.

Particle motion in generalized Dirac’s monopoles of dimension 2k + 1

By using Meng’s idea in his generalization of the classical MICZ-Kepler problem, we obtained the equations of motion of a charged particle in the field of generalized Dirac monopole in odd dimensional Euclidean spaces. The main result is that for every particle trajectory r : I → ℝ2k+1∖{0}, there is a 2-dimensional cone with vertex at the origin on which r is a geodesic.

An obstruction for codimension two contact embeddings in the odd dimensional Euclidean spaces

We prove that the first Chern class of a codimension two closed contact submanifold of the odd dimensional Euclidean space is trivial. For any closed co-oriented contact 3-manifold with trivial Chern class, we prove that there is a contact structure on the 5-dimensional Euclidean space which admits a contact embedding of it.

Intersection homology of linkage spaces in odd-dimensional Euclidean space

We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with $n$ links and prescribed lengths $\ell\in \mathbb{R}^n$ in $d$-dimensional Euclidean space. For $d>3$ these spaces are no longer manifolds generically, but they have the structure of a pseudomanifold. We use intersection homology to assign a ring to these spaces that can be used to distinguish the homeomorphism types of $\mathcal{M}_d(\ell)$ for a large class of length vectors. These rings behave rather differently depending on whether $d$ is even or odd, with the even case having been treated in an earlier paper. The main difference in the odd case comes from an extra generator in the ring which can be thought of as an Euler class of a startified bundle.

On contact embeddings of contact manifolds in the odd dimensional Euclidean spaces

We prove that a closed co-oriented contact (2m + 1)-manifold (M2m + 1, ξ) can be a contact submanifold of the standard contact structure on ℝ4m + 1, if it satisfies one of the following conditions: (1) m is odd (m ≥ 3) and H1(M2m + 1; ℤ) = 0, (2) m is even (m ≥ 4) and M2m + 1 is 2-connected, (3) m = 2 and M5 is simply-connected.

A note on the characteristic rank of null-cobordant manifolds

We give an upper bound for the characteristic rank of smooth closed null-cobordant manifolds and apply it to the Grassmann manifolds of oriented vector subspaces in Euclidean space. The bound turns out to be sharp: it coincides with the exact value of characteristic rank for the Grassmann manifolds of oriented 2-dimensional vector subspaces in odd-dimensional Euclidean space.

Relationship Between Quantum and Poisson Structures of Odd Dimensional Euclidean Spaces

It is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are homeomorphic to the Poisson prime and Poisson primitive spectra of the multiparameter Poisson algebra of odd-dimensional euclidean spaces in the case when the multiplicative subgroup of a base field generated by the parameters is torsion free. As a corollary, it is shown that the prime and primitive spectra of the multiparameter quantized algebra of odd-dimensional euclidean spaces are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring.

Scattering resonances for several small convex bodies and the Lax-Phillips conjecture

This work deals with scattering by obstacles which are finite disjoint unions of strictly convex bodies with smooth boundaries in an odd dimensional Euclidean space. The class of obstacles of this type is considered which are contained in a given (large) ball and have some additional properties: its connected components have bounded eccentricity, the distances between different connected components are bounded from below, and a uniform 'no eclipse condition' is satisfied. It is shown that if an obstacle K in this class has connected components of sufficiently small diameters, then there exists a horizontal strip near the real axis in the complex upper half-plane containing infinitely many scattering resonances (poles of the scattering matrix), i.e. the Modified Lax-Phillips Conjecture holds for such K. This generalizes a well-known result of M. Ikawa concerning balls with the same sufficiently small radius.

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].

Intrinsic equations for the kinematics of a classical vortex string in higher dimensions.

We present here the natural algebraic-geometrical generalization of the Da Rios--Betchov intrinsic equations governing curvature and torsion of an isolated vortex string moving in an unbounded, perfect fluid flow. The filament is embedded in a manifold that can be assumed to be homeomorphic to an odd-dimensional Euclidean space, and whose connection we do not assume to be torsion-free. We suggest how to account for fluid compressibility in the ambient space by its geometrization, and we discuss some special cases of physical interest such as the torsion-free affine connection case and the Riemannian connection case. Finally, we point out the role our results might have in the context of soliton studies.

On normal $(f,\,g,\,u,\,v,\, \lambda)$-structures on submanifolds of codimension $2$ in an even-dimensional Euclidean space

It is well known that a hypersurface of an almost Hermitian manifold admits an almost contact metric structure naturally induced on it. The study of hypersurfaces of a Euclidean space and of a Kahlerian manifold on which the induced almost contact metric structure satisfies certain conditions has been started by one of the present authors [4, 5]. On the other hand Blair [1, 2], Goldberg [3], Ludden [1, 2], Yamaguchi [8] and the present authors [6, 9] started the study of hypersurface of an almost contact manifold and of submanifolds of codimension 2 of an almost complex manifold. These submanifolds admit, under certain conditions, what we call (/, g, u, v, λ) structure. An even-dimensional sphere of codimension 2 of an even-dimensional Euclidean space is a typical example, of a manifold which admits this kind of structure. In a previous paper [9], we have studied the (/, g, u, v, -structure and given characterizations of even-dimensional sphere. In the present paper, we study submanifolds of codimension 2 in an evendimensional Euclidean space which admit a normal (/, g, u, v, Λ)-structure. In § 1, we consider submanifolds of codimension 2 of an even-dimensional Euclidean space regarded as a flat Kahlerian manifold. In the next section, we deal with (/, g, u, v, -structure induced on a submanifold of codimension 2 of an even-dimensional Euclidean space. In § 3, we find differential equations which /, g, u, v and λ satisfy. § 4 is devoted to the study of relations between the structure equations of the submanifold and the induced (/, g, u, v, ^-structure. In § 5 we prove a series of lemmas which are valid for normal (/, g, u, v, λ}structures and in § 6 we study properties of the mean curvature vector of the submanifold with normal (/, g, u, v, -structure. In the last § 7, we study hypersurfaces of an odd-dimensional Euclidean space and determine all the hypersurfaces admitting a normal (/, g, u, v, Λ)-structure. Our main theorem appears at the end of §7.