scholarly journals New homogeneous Einstein metrics on quaternionic Stiefel manifolds

2018 ◽  
Vol 18 (4) ◽  
pp. 509-524 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Homogeneous Space ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Total Space ◽  
Stiefel Manifold ◽  
Isotropy Representation ◽  
Stiefel Manifolds ◽  
Generalized Flag Manifold ◽  
Generalized Wallach Space ◽  
Homogeneous Einstein Metrics

Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

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 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.

scholarly journals New homogeneous Einstein metrics on Stiefel manifolds

2014 ◽  
Vol 35 ◽  
pp. 2-18 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Einstein Metrics ◽  
Stiefel Manifolds ◽  
Homogeneous Einstein Metrics

ON THE NUMBER OF INVARIANT EINSTEIN METRICS ON A COMPACT HOMOGENEOUS SPACE, NEWTON POLYTOPES AND CONTRACTIONS OF LIE ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1047-1075 ◽  
Author(s):  
MICHAIL M. GRAEV
Keyword(s):  
Homogeneous Space ◽  
Lie Algebras ◽  
Einstein Metrics ◽  
Compact Convex ◽  
Convex Polytope ◽  
Invariant Metric ◽  
Isotropy Representation ◽  
Newton Polytopes ◽  
Number Formula ◽  
Proper Face

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number [Formula: see text] of isolate invariant holomorphic Einstein metrics (up to homothety) on Mℂ = Gℂ/Hℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that [Formula: see text], where ν(M) is the integer volume of PM, and give conditions when the defect [Formula: see text]. In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space Mγ = Gγ/HP which is a contraction of M and is associated with a proper face γ of PM.

Invariant Einstein metrics on $\mathrm{SU}(n)$ and complex Stiefel manifolds

2020 ◽  
Vol 72 (2) ◽  
pp. 161-210 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Einstein Metrics ◽  
Stiefel Manifolds
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