Sigurdur Helgason
This paper deals with various topics in analysis on hyperbolic spaces. It surveys some recent progress in non-Euclidean Fourier Analysis and proves some new results for the geodesic Radon transform on hyperbolic spaces.
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In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the $6-$dimensional pseudo-Euclidean space of index two. Also we characterize pseudo-spherical null curves in the $n-$dimensional pseudo-Euclidean space of index two in terms of the curvature functions.
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of GCR hypersurfaces with vanishing Gauss-Kronecker curvature. We also give some explicit examples. Keywords: Generalized constant ratio submanifolds, $δ(r)$-invariant hypersurfaces, constant mean curvature, Gauss-Kronecker curvature
In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3Hrho = Krho^3-2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclide…
This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature and the curvature of the relative metric of a relatively normalized ruled surface $\varPhi$ and in section 3 we introduce some special normalizations of it. All ruled surfaces and their corresponding normalizations that make $\varPhi$ an improper or a proper relative sphere are determined in section 4. In the last section we study ruled surfaces, which are \e…
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by -1. A stronger discrete version was proved by A.D.Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature. Our proof uses "discrete conformal deformations" of the metric that eliminate the singularities and increase…
This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.
Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} to analyze the stability of the differential operator $L_r$ associated with the $r$-th Newton tensor of $f$. This appears in the Jacobi operator for the variational problem of minimizing the $r$-mean curvature $H_r$. Two natural applications are found. The first one ensures that, under the mild condition that the integral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss map meets each equator of $\esse^m$ infinitely many times. The se…
We introduce a class of surfaces in euclidean space motivated by a problem posed by Élie Cartan. This class furnishes what seems to be the first examples of pairs of non-congruent surfaces in euclidean space such that, under a diffeomorphism $Φ$, lines of curvatures are preserved and principal curvatures are switched. We show how to construct such surfaces using holomorphic data and discuss their relation with minimal surfaces. We also prove that if the diffeomorphism $Φ$ preserves the conformal class of the third fundamental form, then all examples belong to the class of surfaces that we deal…
We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal curves, the discrete normals are built up through a Gram-Schmidt procedure applied to consecutive oriented segments, and they naturally live in the projective space associated to the Gauss hyper-sphere. By using sequences of inscribed polygonals with infinitesimal modulus, a relaxed notion of total variation of the $j$-th normal to a generic curve is then introduced. For smooth curves satisfying the Jordan system, in fact, our relaxed notion agre…
We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).
The goal of this article is to investigate nontrivial $m$-quasi-Einstein manifolds globally conformal to an $n$-dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n-1)$-dimensional translation group, we provide a complete classification when $λ=0$ and $m\geq 1$ or $m=2-n.$
We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.
Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity
本文研究了非紧致欧几里得子流形上的对数Sobolev不等式，推导了精确性与刚性的结果，并给出了超收缩性估计的应用。
The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset $ A $ of the Euclidean space and for every integer $ k \geq 2 $ we introduce the approximate differential of order $ k $ of $ A $ and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications.
On the classification of capillary graphs in Euclidean and non-Euclidean spaces
本文研究了欧几里得和非欧几里得空间中具有特定均曲率和边界条件的图的分类问题，通过刚性结果和分裂定理给出了通用的分类方法。
The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the Euclidean space.
Gears and pulleys in non-Euclidean space forms
该研究针对球面几何与双曲几何下的齿轮、滑轮系统展开运动规律分析，发现两类非欧空间中对应的机械运动规律具有有趣的相似性，拓展了经典机械系统的非欧几何研究范畴。
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