Hypersurfaces Of Revolution Research Articles

Generalizing the Equal Area Zones Property of the Sphere

We construct and study n-dimensional hypersurfaces in (n + 1)-dimensional Euclidean space that satisfy a higher dimensional generalization of the equal area zones property of the sphere (that the surface area of a zone between two parallel planes depends only on the distance between the planes). These new hypersurfaces are hypersurfaces of revolution. The relative simplicity of their construction allows us to describe them in great detail, revealing some interesting curiosities and motivating further questions.

HYPERSURFACES WITH POINTWISE 1-TYPE GAUSS MAP

In this paper we prove that an oriented hypersurface $M$ of a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the first kind if and only if $M$ has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of $E^{n+1}$ has 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

HYPERSURFACES WITH POINTWISE 1-TYPE GAUSS MAP

In this paper we prove that an oriented hypersurface $M$ of a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the first kind if and only if $M$ has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of $E^{n+1}$ has 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space $E^{n+1}$ has pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

Cohomogeneity one hypersurfaces of Euclidean Spaces

We study isometric immersions f: M^{n}\to \mathbb{R}^{n+1} into Euclidean space of dimension n+1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either n\geq 3 and M^n is compact or if n\geq 5 and the connected components of the flat part of M^n are bounded. We also provide several sufficient conditions for f to be a hypersurface of revolution.

Generalized Tangential Sphere Bound on the MLDecoding Error Probability of Linear Binary Block Codes in AWGN Interference

The error probability of maximum-likelihood (ML) soft-decision decoded binary block codes rarely accepts nice closed forms. In addition, for long codes, ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system parameters on the overall system performance as well as a measure of goodness of the suboptimum decoding methods used in practice. Using the so-called Gallager's first bounding technique (involving a so-called Gallager region) and within the framework of tangential sphere bound (TSB) of Poltyrev, we develop a general bound referred to as the generalized TSB (GTSB). The Gallager region is chosen to be a general hyper-surface of revolution (HSR) which is optimized to tighten the bound. The search for the optimal Gallager region is a classical problem dating back to Gallager's thesis in the early 1960s. For the random coding case, Gallager provided the optimal solution in a closed form while for the nonrandom case the problem has been an active area of research in information theory for many years. We prove that for a sphere code, the optimal HSR within the proposed GTSB is a hyper-cone. This will climax to the TSB of Poltyrev, one of the tightest bounds ever developed for binary block codes, and therefore terminates the search for a better Gallager region in the groundwork of the GTSB.

Hypersurfaces of cohomogeneity one and hypersurfaces of revolution

In this paper, we study hypersurfaces f :M n→ R n+1, n⩾3 , where M n is a G-cohomogeneity one Riemannian manifold such that the principal orbits of G are umbilical submanifolds of M. In (Ann. Global Anal. Geom. 13 (1995) 169–184), under the assumptions that n⩾4 and M is compact, the authors prove that such a hypersurface must be of revolution. We extend this theorem to the of complete noncompact manifolds, G compact or noncompact. We also study the case n=3.

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\mathbb{R}^n$ (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space.

Constant mean curvature hypersurfaces foliated by spheres

We ask when a constant mean curvature n-submanifold foliated by spheres in one of the Euclidean, hyperbolic and Lorentz–Minkowski spaces ( E n+1 , H n+1 or L n+1 ), is a hypersurface of revolution. In E n+1 and L n+1 we will assume that the spheres lie in parallel hyperplanes and in the case of hyperbolic space H n+1 , the spheres will be contained in parallel horospheres. Finally, Riemann examples in L 3 are constructed, that is, non-rotational spacelike surfaces foliated by circles in parallel planes.

A new result on the Pompeiu problem

A nonernpty bounded open set Q C R' (n > 2) is said to have the Pompeiu property if and only if the only continuous function f on Rn for which the integral of f over oa() is zero for all rigid motions a of Rn is f 0 O. We consider a nonempty bounded open set Q C R'T (n > 2) with Lipschitz boundary and we assume that the complement of Q is connected. We show that the failure of the Pomnpeiu property for Q implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in R', n > 3, has the Pompeiu property. So far the result was proved only for solid tori in JR. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in Rn bounded by hypersurfaces of revolution.

Conformal flatness, cohomogeneity one and hypersurfaces of revolution

In this paper we prove that compact conformally flat cohomogeneity one hypersurfaces of R n + 1 . n ⩾ 4, are hypersurfaces of revolution except for a family of counter-examples that we describe in detail.

Immersions of cohomogeneity one Riemannian manifolds

This work deals with positively curved compact Riemannian manifolds which are acted on by a closed Lie group of isometries whose principal orbits have codimension one and are isotropy irreducible homogeneous spaces. For such manifolds we can show that their universal covering manifold may be isometrically immersed as a hypersurface of revolution in an euclidean space.

Euclidean hypersurfaces with reflection properties

Conic sections and certain hypersurfaces of revolution derived from them are characterized as the only nondegenerate smooth Euclidean hypersurfaces having reflection properties.