Parallel Hypersurfaces Research Articles

Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann boundary to the Dirichlet one is locally the biggest.

Totally Geodesic and Parallel Hypersurfaces of Four-Dimensional Oscillator Groups

We obtain the complete classification and explicitly describe totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, equipped with a one-parameter family of left-invariant Lorentzian metrics.

Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces

We prove that a four-dimensional generalized symmetric space does not admit any non-degenerate hypersurfaces with parallel second fundamental form, in particular non-degenerate totally geodesic hypersurfaces, unless it is locally symmetric. However, spaces which are known as generalized symmetric spaces of type C do admit non-degenerate parallel hypersurfaces and we verify that they are indeed symmetric. We also give a complete and explicit classification of all non-degenerate totally geodesic hypersurfaces of spaces of this type.

ON EXTRINSICALLY SYMMETRIC HYPERSURFACES OF ℍn×ℝ

AbstractTotally umbilical, semi-parallel and parallel hypersurfaces of ℍn×ℝ are completely classified. More examples arise than in the analogous study on the ambient space 𝕊n×ℝ.

On a class of hypersurfaces in % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipC0xg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Wqpe0x % c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr % 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1 % uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-jj8tbaa!456F! $$ \mathbb{S} $$ n × ℝ and ℍ n × ℝ

We give a complete description of all hypersurfaces of the product spaces \( \mathbb{S} \)n × ℝ and ℍn × ℝ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces ℝn+2 ⊃ \( \mathbb{S} \)n × ℝ and \( \mathbb{L} \)n+2 ⊃ ℍn × ℝ. We prove that any such hypersurface in \( \mathbb{S} \)n × ℝ (respectively, ℍn × ℝ) can be constructed by means of a family of parallel hypersurfaces in \( \mathbb{S} \)n (respectively, ℍn) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class correspond to an isoparametric family in the base space and a smooth function that is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of \( \mathbb{S} \)n × ℝ and ℍn × ℝ, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor ℝ. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in \( \mathbb{S} \)n × ℝ and ℍn × ℝ Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.

The global structure of locally convex hypersurfaces in Finsler—Hadamard manifolds

Locally convex compact immersed hypersurfaces in the Finsler—Hadamard space with bounded T-curvature are considered. Under certain conditions on normal curvatures, such hypersurfaces are proved to be convex, embedded, and homeomorphic to the sphere. To this end, the Rauch theorem is generalized to exponential maps of hypersurfaces and the convexity of parallel hypersurfaces is proved.

Orthogonal curvature of parallel hypersurfaces

In this paper we obtain the orthogonal curvature of parallel hypersurfaces using Euler’s and some preliminaries theorem then we show if the parallel map preserves the fundamental 2-form then the hypersurfaces will be hyperplanes.

A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres

The algorithm proved here solves the problem of orthogonal distance regression for the maximum norm with hyperplanes and hyperspheres. For each finite set of points in a Euclidean space of any dimension, the algorithm determines --- through finitely many arithmetic operations --- all the hyperplanes and hyperspheres that minimize the maximum Euclidean distance measured perpendicularly from the data. The algorithm finds all the slabs (bounded by parallel hyperplanes) and all the spherical shells (bounded by concentric hyperspheres) that contain all the data and are "rigidly supported" by the data (for which there does not exist any other pair of parallel hypersurfaces of the same type that intersect the data at the same points.) The computational complexity of the algorithm increases as the number of data points raised to the dimension of the ambient space. The solutions are then the midrange hyperplanes in the thinnest slabs, and the midrange hyperspheres in the thinnest shells. Their sensitivity to perturbations of the data is of the order of a power of the reciprocal of the smallest angle between two median hyperplanes separating two pairs of data points. The methods of proof consist in showing that if a pair of parallel hyperplanes or hyperspheres is not rigidly supported but encompasses all the data, then there exists a projective shift of their common projective center producing a thinner slab or shell that still contains all the data.

Comparison and rigidity theorems in semi-Riemannian geometry

The comparison theory for the Riccati equation satis- ed by the shape operator of parallel hypersurfaces is generalized to semi{Riemannian manifolds of arbitrary index, using one{sided bounds on the Riemann tensor which in the Riemannian case cor- respond to one{sided bounds on the sectional curvatures. Starting from 2{dimensional rigidity results and using an inductive tech- nique, a new class of gap{type rigidity theorems is proved for semi{ Riemannian manifolds of arbitrary index, generalizing those rst given by Gromov and Greene{Wu. As applications we prove rigid- ity results for semi{Riemannian manifolds with simply connected ends of constant curvature.

Higher order gaussian curvatures of parallel hypersurfaces

Let M be a hypersutface in (n+1) -dimensional Euclidean space En+1 and M be a parallel hypersutface to M.

On the curvates of the parallel hypersurfaces

On the curvates of the parallel hypersurfaces

On the parallel hypersurfaces with constant curvature

Gaussian and mean curvatures, K and for parallel surfaces in E3 are given in [2 ]. In the present note, by means of higher order Gaussian and mean curvatures, we calculate the generaiized the curvatures Kr and Hr in En+1, n > 2

Affine higher order parallel hypersurfaces

Affine higher order parallel hypersurfaces

On the Gaussian and mean curvatures of parallel hypersurfaces

In this study, we give a generaiizatîon of the higher order Gaussian curvature function and the mean curvature function ahout the parallel surfaces in E^, for E^’^*.

Comparison theorems and hypersurfaces

We compare the second fundamental forms of a family of parallel hypersurfaces in different Riemannian manifolds. This leads to new proofs for the distance and volume comparison theorems in Riemannian geometry. In particular, we get a new result on the volume of the set of points with distance≤r from a totally geodesic submanifold, for any r. The analytic prerequisite is the investigation of the Riccati type ODE which is satisfied by the second fundamental form of a parallel hypersurface family.

The theory of integration on manifold and the volume for parallel hypersurfaces

We generalized the theorem about the hyper ar e a element dA for the manifold of orient u in En and by using this theorem we proved the generalized Divergence Theorem. First we defined the outward unit normal vector. filed of manifold M in a hypersurface as:

Applications of stereographic projections to submanifolds in 𝐸^{𝑚} and 𝑆^{𝑚}

In this paper we give a criterion for a compact minimal submanifold of S m {S^m} to lie in a given great hypersphere in terms of an integral over the stereographic image in E m {E^m} of the submanifold. We also show that if all the points a certain normal distance C C from a compact hypersurface M M in E m {E^m} lie on a sphere of radius D > C D > C then M M is a hypersphere. This generalizes a classical result on parallel hypersurfaces. We prove this theorem by showing it to be equivalent, via stereographic projection, to a recent result of Nomizu and Smyth concerning the gauss map for hypersurfaces of S m {S^m} .

On parallel hypersurfaces of an elliptic hypersurface of second order in $E^{n+1}$

On parallel hypersurfaces of an elliptic hypersurface of second order in $E^{n+1}$

On Parallel Hypersurfaces in the Elliptic and Hyperbolic n-Dimensional Space

On Parallel Hypersurfaces in the Elliptic and Hyperbolic n-Dimensional Space

On parallel hypersurfaces in the elliptic and hyperbolic 𝑛-dimensional space

On parallel hypersurfaces in the elliptic and hyperbolic 𝑛-dimensional space