Hypersurfaces with Isometric Reeb Flow in Hermitian Symmetric Spaces of Rank 2

2013 ◽  
pp. 293-301
Author(s):  
Young Jin Suh
Keyword(s):  
Symmetric Spaces ◽  
Hermitian Symmetric Spaces ◽  
Rank 2

Cyclic parallel hypersurfaces in complex Grassmannians of rank 2

2019 ◽  
Vol 31 (02) ◽  
pp. 2050014 ◽  
Author(s):  
Hyunjin Lee ◽  
Young Jin Suh
Keyword(s):  
Vector Field ◽  
Geometric Structure ◽  
Symmetric Spaces ◽  
Shape Operator ◽  
Hermitian Symmetric Spaces ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Hopf Hypersurfaces ◽  
Rank 2 ◽  
Parallel Hypersurfaces

The object of the paper is to study cyclic parallel hypersurfaces in complex (hyperbolic) two-plane Grassmannians which have a remarkable geometric structure as Hermitian symmetric spaces of rank 2. First, we prove that if the Reeb vector field belongs to the orthogonal complement of the maximal quaternionic subbundle, then the shape operator of a cyclic parallel hypersurface in complex hyperbolic two-plane Grassmannians is Reeb parallel. By using this fact, we classify all cyclic parallel hypersurfaces in complex hyperbolic two-plane Grassmannians with non-vanishing geodesic Reeb flow. Next, we give a non-existence theorem for cyclic Hopf hypersurfaces in complex two-plane Grassmannians.

scholarly journals ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

1996 ◽  
Vol 11 (18) ◽  
pp. 3257-3295 ◽  
Author(s):  
F. TOPPAN
Keyword(s):  
Power Series ◽  
Symmetric Spaces ◽  
Unexpected Result ◽  
Hermitian Symmetric Spaces ◽  
Regular Elements ◽  
Symmetry Properties ◽  
Alternating Sums ◽  
Spectral Parameter Power Series ◽  
Special Limit ◽  
Free Parameters

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

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