On Peterson surfaces of constant Gaussian curvature

1990 ◽  
Vol 51 (2) ◽  
pp. 2189-2190
Author(s):  
Ya. P. Blank ◽  
N. M. Gormashova
Keyword(s):  
Gaussian Curvature ◽  
Constant Gaussian Curvature

scholarly journals Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

2021 ◽  
Author(s):  
Wojciech Szumiński ◽  
Andrzej J. Maciejewski
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Curvature ◽  
Necessary Conditions ◽  
Constant Gaussian Curvature ◽  
Local Coordinates ◽  
Hamiltonian Functions

AbstractIn the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

scholarly journals Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 72
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri
Keyword(s):  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Two Dimensional ◽  
Constant Gaussian Curvature ◽  
Metric Structures ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Full Classification ◽  
Kaluza Klein

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

An Invariant Membrane Stress Function for Shells

1953 ◽  
Vol 20 (2) ◽  
pp. 178-182
Author(s):  
H. L. Langhaar
Keyword(s):  
Gaussian Curvature ◽  
Stress Function ◽  
Airy Function ◽  
Membrane Stress ◽  
Constant Gaussian Curvature ◽  
Bending Stresses ◽  
Small Deformations

Abstract Inextensional shells that have no thickness are idealized representations of real shells that have small bending stresses and small deformations. With certain restrictions, the stresses in these shells are derivable from a generalized Airy function. For shells of constant Gaussian curvature, the stress function is unrestricted, but, for other shells, it is expressed as a function of the Gaussian curvature. Although, in this respect, it is less general than Pucher’s stress function, it has the advantage that it may be used with any surface co-ordinates.

scholarly journals Encoding Gaussian curvature in glassy and elastomeric liquid crystal solids

2016 ◽  
Vol 472 (2189) ◽  
pp. 20160112 ◽  
Author(s):  
Cyrus Mostajeran ◽  
Mark Warner ◽  
Taylor H. Ware ◽  
Timothy J. White
Keyword(s):  
Gaussian Curvature ◽  
Logarithmic Spiral ◽  
Geometric Constraints ◽  
Intrinsic Geometry ◽  
Constant Gaussian Curvature ◽  
Spatially Inhomogeneous ◽  
Negative Gaussian Curvature ◽  
Annular Domains ◽  
Spherical Caps ◽  
Local Deformations

We describe shape transitions of thin, solid nematic sheets with smooth, preprogrammed, in-plane director fields patterned across the surface causing spatially inhomogeneous local deformations. A metric description of the local deformations is used to study the intrinsic geometry of the resulting surfaces upon exposure to stimuli such as light and heat. We highlight specific patterns that encode constant Gaussian curvature of prescribed sign and magnitude. We present the first experimental results for such programmed solids, and they qualitatively support theory for both positive and negative Gaussian curvature morphing from flat sheets on stimulation by light or heat. We review logarithmic spiral patterns that generate cone/anti-cone surfaces, and introduce spiral director fields that encode non-localized positive and negative Gaussian curvature on punctured discs, including spherical caps and spherical spindles. Conditions are derived where these cap-like, photomechanically responsive regions can be anchored in inert substrates by designing solutions that ensure compatibility with the geometric constraints imposed by the surrounding media. This integration of such materials is a precondition for their exploitation in new devices. Finally, we consider the radial extension of such director fields to larger sheets using nematic textures defined on annular domains.

Hamiltonian formulation of surfaces with constant Gaussian curvature

2009 ◽  
Vol 42 (42) ◽  
pp. 425204 ◽  
Author(s):  
Miguel Trejo ◽  
Martine Ben Amar ◽  
Martin Michael Müller
Keyword(s):  
Gaussian Curvature ◽  
Hamiltonian Formulation ◽  
Constant Gaussian Curvature

scholarly journals Totally real surfaces inCP2with parallel mean curvature vector

1992 ◽  
Vol 15 (3) ◽  
pp. 589-592
Author(s):  
M. A. Al-Gwaiz ◽  
Sharief Deshmukh
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Real Surface ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Constant Gaussian Curvature ◽  
Totally Geodesic ◽  
Parallel Mean Curvature ◽  
Totally Real

It has been shown that a totally real surface inCP2with parallel mean curvature vector and constant Gaussian curvature is either flat or totally geodesic.

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