Euclidean Sphere Research Articles

A Sphere Theorem for Reverse Volume Pinching on Even-Dimensional Manifolds

Let $M$ be a compact simply connected riemannian manifold of even dimension $d$. It is well known that if the sectional curvature of $M$ lies in the range $\left ( {0,\lambda } \right ]$, then $M$ has volume greater than or equal to that of the $d$-dimensional euclidean sphere $S_\lambda ^d$ of constant curvature $\lambda$. We prove that if the volume of $M$ is no greater than 3/2 times that of $S_\lambda ^d$, then $M$ is homeomorphic with the sphere.

The construction of analytic diffeomorphisms for exact robot navigation on star worlds

A Euclidean Sphere World is a compact connected submanifold of Euclidean n n -space whose boundary is the disjoint union of a finite number of ( n − 1 ) (n - 1) dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World "model." Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results.

Plücker formulae for the orthogonal group

Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

Sur une caracterisation de la sphereS 2n+1 en termes de geometrie riemannienne de contact

One proves that the unique hypersurfacesS⊂IR2n such that a contact structure is isohelicoidal are the Euclidean spheres centered at the origin.

Approximations of Sobolev maps between an open set and an Euclidean sphere, boundary data, and singularities

The problem of density of smooth maps between manifolds was posed in [EL] and in [SU]. In [BZ], Bethuel and Zheng showed that smooth maps from an open subset 12 of R ~ are dense in WI'P(O, S ~ 1) if 1 < p < M 1. Here we extend this result to approximate maps of W 1' P(f2, S M1) which have some prescribed smooth boundary data by maps with the same boundary data, and which are smooth on fl excepted on a closed subset 6 a with zero p-capacity, still if 1 ~ p < M 1. The first part of this paper is devoted to the basic tools used in the proof of this result. In the second part the proof of our main theorem is given; more precisely: We consider p a real number in [1, + ~ ) and two integer N and M larger than 1. S ~ 1 = {y ~ RM/Iy] = 1 } is the unit sphere of R M. 12 is a bounded open set of R N, whose boundary dr2 is a smooth (N-1)-dimensional compact manifold. Let f be in WI'P(f2,SU-1), we define

Constant isotropic submanifolds with 4-planar geodesics

Let f f be an isometric immersion of a Riemannian manifold M M into M ¯ \overline M . We prove that if f f is constant isotropic, 4 4 -planar geodesic and M ¯ \overline M is a Euclidean sphere, then M M is isometric to one of compact symmetric spaces of rank equal to one and f f is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic, 4 4 -planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature.

The low-dimensional metric foliations of Euclidean spheres

The low-dimensional metric foliations of Euclidean spheres

Embeddings of Ultrametric Spaces in Finite Dimensional Structures

Motivated by recent advances in theoretical physics and combinatorial optimization, we study the problem of embedding ultrametric spaces into finite dimensional structures: finite sets, Euclidean spaces $\mathbb{R}^n $, Euclidean sphere $S^n $, and n-dimensional hypercube with Hamming distance. We give conditions and constructions of embeddings and show a general upper bound of $n + 1$ on the cardinality of the ultrametric set. We also give an upper bound on the cardinality of quasi-ultrametric sets.

Zero-Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients

We prove that the zero set of any quaternionic (or octonionic) analytic function f with central (that is, real) coefficients is the disjoint union of codimension two spheres in R4 or R8 (respectively) and certain purely real points. In particular, for polynomials with real coefficients, the complete root-set is geometrically characterisable from the lay-out of the roots in the complex plane. The root-set becomes the union of a finite number of codimension 2 Euclidean spheres together with a finite number of real points. We also find the preimages f−1 for any quaternion (or octonion) A. We demonstrate that this surprising phenomenon of complete spheres being part of the solution set is very markedly a special ‘real’ phenomenon. For example, the quaternionic or octonionic Nth roots of any non-real quaternion (respectively octonion) turn out to be precisely N distinct points. All this allows us to do some interesting topology for self-maps of spheres.

Harmonic Maps and a Pinching Theorem for Positively Curved Hypersurfaces

In this paper, we establish a theorem of Liouville type for stable harmonic maps in sufficiently pinched, positively curved hypersurfaces of a space form with nonnegative constant curvature. Similar results for the Euclidean sphere ${S^n}$ have been proved by Y. L. Xin and P. F. Leung, respectively.

R-spaces associated with a Hermitian symmetric pair

The linear isotropy representation of a Riemannian symmetric pair (G,K) is defined as the differentialof the left action of K on GJK at the origin. Every orbit of the linear isotropy representation of (G,K) is called an R-space associated with {G,K), which is an important example of equivariant homogeneous Riemannian submanifolds in a Euclidean sphere (See Takagi-Takahashi [2] and TakeuchiKobayashi [3]). This paper is concerned with the linear isotropy representation of a Hermitian symmetric pair (G,K). Its restriction to the center of K defines an S1-action on the associated i?-spaces. We determine all i?-spaces associated with Hermitian symmetric pairs (G,K) on which the semisimple part of K acts transitively. In particular,we know allirreducible Hermitian symmetric pairs such that each of the associated i?-spaces has such a property. This result is utilizablefor the classificationof orthogonal transformation groups by their cohomogeneity (See the forthcoming paper [4] concerned with this problem in low cohomogeneity). The authors are profoundly grateful to Professor Ryoichi Takagi for his helpful suggestion and criticalreading of a primary manuscript.

Isotropic Immersions

Recently, Ferus ([5], [6]) classified connected Riemannian manifolds with parallel second fundamental form in a real space form of constant curvature . In this paper we may restrict our attention to isotropic submanifolds with parallel second fundamental form in the Euclidean sphere Sm(k) of constant curvature k. Due to Ferus, we find that an isotropic submanifold with parallel second fundamental form in Sm is locally congruent to one of compact symmetric spaces of rank one and the immersion is locally equivalent to the second or the first standard immersion according as M is a sphere or not. In Section 2, we characterize the first standard immersion of a complex projective space into a sphere in terms of isotropic immersions.

Zur kinematik isogonaler gleitkurven

Pairs of curves are discussed during a motion in the euclidean plane or on a euclidean sphere, which are crossing themselves with a constant angle α ≠ 0 during the whole motion. These curves are a generalization of associated envelopes ( α = 0) and can be characterized by complex roll-sliding numbers. Some of the well known properties of associated envelopes can be extended to these associated isogonal sliding curves.

Sequencing Problems in Two-Server Systems

We analyze a service system in which two identical servers move one at a time along a linear array of N positions. Requests for service, each designating one of the N positions, join a first-in-first-out queue, where processing of the nth request does not begin until processing of the (n − 1)th requested is completed. Processing the nth request entails determining which server to move, moving this server to the requested position, and then performing the service. Several potential applications of the model are mentioned, the most notable being the design of computer storage systems with multiple access devices. Within a simple probability model we compare server-selection policies in terms of the equilibrium expected distance a server is moved in processing a request. Distance is measured under two regimes, both assigning a unit distance between adjacent positions. In the first, the positions are thought of as equally spaced along a line (interval), and in the second, equally spaced along a circle (positions 1 and N are adjacent). For the interval problem the server-selection policies examined include the partition rule, whereby die interval is bisected and a server restricted to the requests in each half-interval, and the nearer-server rule whereby service is always provided by the server closer to the requested position, with ties broken in favor of the right-hand server. Under the assumption that requested positions are distributed uniformly over the set of N, our results show that expected server motion under the nearer-server rule is approximately 0.1625 N for moderately large N, which is to be compared with the known result of N/6 = 0.1666 … N for the partition rule. We also derive an optimization rule for which numerical calculations indicate that 0.1598 N is the approximate performance for large N. A number of other results plus two conjectures are presented which characterize optimization rules on the interval. Finally, extensive numerical calculations are displayed which describe the joint distributions of server positions under the nearer-server and a related rule. For the more tractable circle problem we prove that die nearer-server rule is optimal, and we derive an explicit form for the stationary density of the distance between servers. It is noted that the nearer-server rule is also optimal when the circle is generalized to 2-point homogeneous spaces (e.g., a Euclidean sphere) in more than one dimension.

On the construction of codimension two minimal immersions of exotic spheres into Euclidean spheres

It is, nowadays, a well-known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non-diffeomorphic to spheres. Such manifolds were christened exotic spheres by their founder, Milnor [223. The structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in [20]. For example, it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres. In differential geometry and geometric variational theory, the study of closed minimal submanifolds of the euclidean n-sphere, S"(1), is directly related to that of the local structure of singularities of minimal subvarieties in the general Riemannian setting. Therefore, closed minimal submanifolds of S"(1) are not only interesting, nice, geometric objects by themselves, but are also of general theoretical importance in the study of geometric measure theory [16, 18, 24]. Among various problems on the closed minimal submanifolds of euclidean spheres, the following two problems naturally distinguish themselves as especially interesting, namely:

Homogeneous Minimal Surfaces in Euclidean Spheres with Flat Normal Connections

Homogeneous Minimal Surfaces in Euclidean Spheres with Flat Normal Connections

De Sitter-invariant vacuum state and temperature green function

Using the method of canonical quantization in the static spacetime, we calculated the temperature Green function of the real scalar field of the de Sitter spacetime. We then used the generating functional of the Green function in the euclidean path integral representation to prove that the temperature Green function for T = ( Λ 3 ) 2π , Λ being positive cosmological constant, is identical with the Green function G E on the 4-dimensional euclidean sphere, thereby showing that the de Sitter-invariant vacuum state with respect to an inertial measuring system (geodesic observer) is the quantum mixed state with a Hawking temperature equal to T.

Homogeneous minimal surfaces in Euclidean spheres with flat normal connections

We classify, up to congruence, homogeneous minimal surfaces in Euclidean spheres with flat normal connections. The parameter varieties in the space of contact invariants of E. Cartan are computed for all codimensions.

Geometric quantization of the multidimensional Kepler problem

The geometric quantization scheme of Czyz and Hess is applied to the (n - 1)-dimensional quadric in complex projective space. As the quadratic is the orbit manifold of the n-dimensional Kepler problem and the geodesic flow on the n-dimensional euclidean sphere, we thus obtain the quantum energy levels and their multiplicities for these Hamiltonian systems.

An addition formula for hyperbolic space

The addition formula for Gegenbauer polynomials on the Euclidean sphere is generalized to the unit hyperboloid. Applications are given in terms of new bounds on the cardinality of few-distance sets in Lobatchevski space.