Equifocal submanifolds in a symmetric space and the infinite dimensional geometry

2019 ◽  
Vol 32 (1) ◽  
pp. 25-56
Author(s):  
Naoyuki Koike
Keyword(s):  
Symmetric Space ◽  
Infinite Dimensional ◽  
Equifocal Submanifolds

scholarly journals PROPER CURVATURE COLLINEATIONS IN NONSTATIC SPHERICALLY SYMMETRIC SPACE–TIMES

2008 ◽  
Vol 23 (05) ◽  
pp. 749-759 ◽  
Author(s):  
GHULAM SHABBIR ◽  
M. RAMZAN
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Infinite Dimensional ◽  
Riemann Matrix ◽  
Integration Techniques ◽  
Curvature Collineations ◽  
Proper Curvature

A study of nonstatic spherically symmetric space–times according to their proper curvature collineations is given by using the rank of the 6×6 Riemann matrix and direct integration techniques. Studying proper curvature collineations in each case of the above space–times it is shown that when the above space–times admit proper curvature collineations, they turn out to be static spherically symmetric and form an infinite dimensional vector space. In the nonstatic cases curvature collineations are just Killing vector fields.

scholarly journals G -Strands on symmetric spaces

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160795 ◽  
Author(s):  
Alexis Arnaudon ◽  
Darryl D. Holm ◽  
Rossen I. Ivanov
Keyword(s):  
Symmetric Space ◽  
Lie Group ◽  
Particle Physics ◽  
Symmetric Spaces ◽  
Sobolev Norm ◽  
Chiral Model ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Broken Symmetries ◽  
Dimensional Group

We study the G -strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G , and we treat in more detail examples with symmetric space SU (2)/ S 1 and SO (4)/ SO (3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G -strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa–Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.

scholarly journals On the transverse Scalar Curvature of a Compact Sasaki Manifold

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Weiyong He
Keyword(s):  
Symmetric Space ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Moment Map ◽  
Sasaki Metric ◽  
Kahler Geometry ◽  
Infinite Dimensional ◽  
Standard Picture ◽  
Constant Scalar Curvature Metrics ◽  
Sasaki Manifold

AbstractWe show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group

scholarly journals SYMMETRIES OF LOCALLY ROTATIONALLY SYMMETRIC MODELS

2005 ◽  
Vol 14 (10) ◽  
pp. 1675-1684 ◽  
Author(s):  
M. SHARIF
Keyword(s):  
Symmetric Space ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Interesting Case ◽  
Degenerate Case ◽  
Infinite Dimensional ◽  
Constraint Equations ◽  
Matter Collineations ◽  
Locally Rotationally Symmetric ◽  
Rotationally Symmetric

Matter collineations of locally rotationally symmetric space–times are considered. These are investigated when the energy–momentum tensor is degenerate. We know that the degenerate case provides infinite dimensional matter collineations in most of the cases. However, an interesting case arises where we obtain proper matter collineations. We also solve the constraint equations for a particular case to obtain some cosmological models.

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