scholarly journals QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE

2010 ◽  
Vol 81 (3) ◽  
pp. 496-506 ◽  
Author(s):  
JOSU ARROYO ◽  
ÓSCAR J. GARAY ◽  
JOSE MENCÍA
Keyword(s):  
Variational Analysis ◽  
Lagrange Equation ◽  
Numerical Approach ◽  
Real Space ◽  
Closed Curves ◽  
Space Forms ◽  
Energy Functionals ◽  
Critical Curves ◽  
Real Space Forms ◽  
Curvature Energy

AbstractThe classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.

RELATIVISTIC PARTICLES ALONG NULL CURVES IN 3D LORENTZIAN SPACE FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2851-2859 ◽  
Author(s):  
ÁNGEL GIMÉNEZ
Keyword(s):  
Vector Fields ◽  
Lagrange Equation ◽  
Killing Vector ◽  
Space Forms ◽  
Lorentzian Space ◽  
Motion Equations ◽  
Critical Curves ◽  
Null Curves ◽  
Cartan Curvature ◽  
Relativistic Particles

We study relativistic particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in (2 + 1)-dimensions backgrounds with constant curvature. We obtain first integrals of the Euler–Lagrange equation by using geometrical methods involving the search for Killing vector fields along critical curves of the action. In the case in which Lagrangian density depends quadratically on Cartan curvature, it is shown that the mechanical system is governed by a stationary Korteweg–De Vries system. Motion equations are completely integrated by quadratures in terms of elliptic and hyperelliptic functions.

scholarly journals A Class of Special Hypersurfaces in Real Space Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yan Zhao ◽  
Ximin Liu
Keyword(s):  
Real Space ◽  
Principal Curvatures ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Constant Principal Curvatures

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

scholarly journals A new characterization of complete linear Weingarten hypersurfaces in real space forms

2013 ◽  
Vol 261 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Cícero Aquino ◽  
Henrique de Lima ◽  
Marco Velásquez
Keyword(s):  
Real Space ◽  
Space Forms ◽  
Linear Weingarten Hypersurfaces ◽  
Real Space Forms

scholarly journals A fundamental theorem for submanifolds of multiproducts of real space forms

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Marie-Amélie Lawn ◽  
Julien Roth
Keyword(s):  
Minimal Surfaces ◽  
Fundamental Theorem ◽  
Arbitrary Number ◽  
Real Space ◽  
Isometric Immersions ◽  
Space Forms ◽  
Real Space Forms ◽  
Simply Connected ◽  
Bonnet Theorem

AbstractWe prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊

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