On the Relation between Real Euclidean and Complex Projective Geometry

1961 ◽  
Vol 45 (352) ◽  
pp. 108 ◽  
Author(s):  
E. C. Zeeman
Keyword(s):  
Projective Geometry ◽  
Complex Projective

PHASE OSCILLATORS WITH SINUSOIDAL COUPLING INTERPRETED IN TERMS OF PROJECTIVE GEOMETRY

2011 ◽  
Vol 21 (06) ◽  
pp. 1795-1804 ◽  
Author(s):  
IAN STEWART
Keyword(s):  
Coordinate System ◽  
Symmetry Group ◽  
Projective Geometry ◽  
Three Dimensional ◽  
Orbit Space ◽  
Alternative Proof ◽  
Phase Oscillators ◽  
Homogeneous Coordinates ◽  
Möbius Group ◽  
Complex Projective

Marvel et al. [2009] studied sinusoidally coupled phase oscillators, generalizing coupled Josephson junctions. They obtained an explicit reduction of the dynamics to a parametrised family of ODEs on the three-dimensional Möbius group. This differs from the usual reduction on to the orbit space of a symmetry group. We apply the viewpoint of complex projective geometry to obtain an alternative proof that trajectories lie on orbits of the Möbius group, and derive a different explicit form for the reduced ODE. The main innovation is the use of homogeneous coordinates, which linearize the action of the Möbius group and lead to a simple coordinate system in which to write the reduced ODE. We also discuss a Lie-theoretic interpretation.

On submersions of the complex indicatrix

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5081-5092
Author(s):  
Elena Popovicia
Keyword(s):  
Fixed Point ◽  
Complex Projective Space ◽  
Finsler Space ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Codimension One ◽  
Real Submanifold ◽  
Fundamental Equations ◽  
Complex Projective ◽  
Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

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