scholarly journals The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

2011 ◽  
Vol 61 (8) ◽  
pp. 1587-1600 ◽  
Author(s):  
Stavros Anastassiou ◽  
Ioannis Chrysikos
Keyword(s):  
Ricci Flow ◽  
Einstein Metrics ◽  
Flag Manifolds ◽  
Homogeneous Einstein Metrics

scholarly journals HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS

2013 ◽  
Vol 24 (10) ◽  
pp. 1350077 ◽  
Author(s):  
ANDREAS ARVANITOYEORGOS ◽  
IOANNIS CHRYSIKOS ◽  
YUSUKE SAKANE
Keyword(s):  
Einstein Equation ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Polynomial Systems ◽  
Flag Manifolds ◽  
Isotropy Representation ◽  
Generalized Flag Manifolds ◽  
A New Technique ◽  
Kähler Einstein Metrics ◽  
Homogeneous Einstein Metrics

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

scholarly journals Proving isometry for homogeneous Einstein metrics on flag manifolds by symbolic computation

2013 ◽  
Vol 55 ◽  
pp. 59-71 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Ioannis Chrysikos ◽  
Yusuke Sakane
Keyword(s):  
Symbolic Computation ◽  
Einstein Metrics ◽  
Flag Manifolds ◽  
Homogeneous Einstein Metrics

scholarly journals INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS

2011 ◽  
Vol 90 (2) ◽  
pp. 237-251 ◽  
Author(s):  
ANDREAS ARVANITOYEORGOS ◽  
IOANNIS CHRYSIKOS
Keyword(s):  
Variational Approach ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Semisimple Lie Group ◽  
Generalized Flag Manifolds ◽  
Generalized Flag Manifold ◽  
Scalar Curvature Functional ◽  
Homogeneous Einstein Metrics ◽  
Adjoint Orbit

AbstractLet M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

2011 ◽  
Vol 10 (2) ◽  
pp. 307-325 ◽  
Author(s):  
Caio J. C. Negreiros ◽  
Lino Grama ◽  
Neiton P. da Silva
Keyword(s):  
Einstein Metrics ◽  
Harmonic Maps ◽  
Flag Manifolds
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