scholarly journals A class of curvature flows expanded by support function and curvature function in the Euclidean space and hyperbolic space

2021 ◽  
pp. 109305
Author(s):  
Shanwei Ding ◽  
Guanghan Li
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Support Function ◽  
Curvature Function

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

Holomorphic Mappings of the Hyperbolic Space into the Complex Euclidean Space and the Bloch Theorem

1975 ◽  
Vol 27 (2) ◽  
pp. 446-458 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Hyperbolic Space ◽  
Holomorphic Mapping ◽  
Continuous Functions ◽  
Holomorphic Mappings ◽  
Euclidean Metric ◽  
The Family ◽  
Complex Euclidean Space ◽  
Bounded Holomorphic

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

scholarly journals An integral formula for compact hypersurfaces in a Euclidean space and its applications

1992 ◽  
Vol 34 (3) ◽  
pp. 309-311 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Smooth Function ◽  
Support Function ◽  
Integral Formula ◽  
Position Vector ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field

Let M be a compact hypersurface in a Euclidena space ℝn+1. The support function p of M is the component of the position vector field of Min ℝn+1 along the unit normal vector field to M, which is a smooth function defined on M. Let S be the scalar curvature of M. The object of the present paper is to prove the following theorems.

Evolving convex surfaces to constant width ones

2017 ◽  
Vol 28 (11) ◽  
pp. 1750082
Author(s):  
Laiyuan Gao ◽  
Yuntao Zhang
Keyword(s):  
Euclidean Space ◽  
Spherical Harmonic ◽  
Support Function ◽  
Convex Surface ◽  
Constant Width ◽  
Evolution Model ◽  
Initial Surface ◽  
Principal Curvatures ◽  
Convex Surfaces ◽  
Evolving Surface

Given an [Formula: see text]-dimensional convex surface [Formula: see text] in the Euclidean space [Formula: see text], this initial surface can be deformed into a convex surface with constant width by a new evolution model which preserves the convexity of the evolving surface, provided that the initial principal curvatures satisfy a [Formula: see text]-pinching condition. Some examples of the flow are also constructed via spherical harmonic expansion of the support function.

scholarly journals Rigid Body Trajectories in Different 6D Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Carol Linton ◽  
William Holderbaum ◽  
James Biggs
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Hyperbolic Space ◽  
Poisson Structure ◽  
Scaling Factor ◽  
The Body ◽  
Body Frame ◽  
Spatial Frame ◽  
Axis Of Symmetry ◽  
Rotational Motions

The objective of this paper is to show that the group with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

scholarly journals Distance functions and umbilic submanifolds of hyperbolic space

1979 ◽  
Vol 74 ◽  
pp. 67-75 ◽  
Author(s):  
Thomas E. Cecil ◽  
Patrick J. Ryan
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Hyperbolic Space ◽  
Critical Points ◽  
Morse Function ◽  
Complete Riemannian Manifold ◽  
Distance Functions

In 1972, Nomizu and Rodriguez [5] found the following characterization of the complete umbilic submanifolds of Euclidean space.Theorem A. Let Mn, n ≥ 2, be a connected, complete Riemannian manifold isometrically immersed in a Euclidean space Em. Every Morse function of the form Lp has index 0 or n at all of its critical points if and only if Mnis embedded as a Euclidean n-subspace or a Euclidean n-sphere in Em.

Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200372
Author(s):  
Stephen T. Hyde ◽  
Martin Cramer Pedersen
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Two Dimensional ◽  
Hyperbolic Surfaces ◽  
Euclidean Three Space ◽  
Three Space ◽  
Planar Graphene

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n  = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

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