Some New Homogeneous Einstein Metrics on Symmetric Spaces

Kerr, Megan M.

doi:10.1090/s0002-9947-96-01512-7

Cited by 28 publications

(23 citation statements)

References 11 publications

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“…for some x 1 > 0 and x 2 > 0. Note, that the subset of SO(2n)-invariant (symmetric) metrics on G/H consists of the metrics with the relation x 2 = 2x 1 [26]. As in the previous case, every SO(2n − 1)-invariant metric on SO(2n − 1)/U (n − 1) is weakly symmetric and, hence, g.o.…”

Section: Theorem 23 Implies Easily the Classification Of Transitive A...mentioning

confidence: 87%

On δ-homogeneous Riemannian manifolds

Berestovskiî¹,

Nikonorov²

2008

Differential Geometry and its Applications

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We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any δ-homogeneous Riemannian manifolds is itself δ-homogeneous. In turn, every simply connected Riemannian δ-homogeneous manifold is a direct metric product of an Euclidean space and compact simply connected indecomposable homogeneous manifolds; all factors in this product are itself δ-homogeneous. We find different characterizations of δ-homogeneous Riemannian spaces, which imply that any such space is geodesic orbit (g.o.) and every normal homogeneous Riemannian manifold is δ-homogeneous. The g.o. property and the δ-homogeneity property are inherited by closed totally geodesic submanifolds. Then we find all possible candidates for compact simply connected indecomposable Riemannian δ-homogeneous non-normal manifolds of positive Euler characteristic and a priori inequalities for parameters of the corresponding family of Riemannian δ-homogeneous metrics on them (necessarily two-parametric). We prove that there are only two families of possible candidates: non-normal (generalized) flag manifolds SO(2l +1)/U (l) and Sp(l)/U (1)·Sp(l −1), l ≥ 2, investigated earlier by W. Ziller, H. Tamaru, D.V. Alekseevsky and A. Arvanitoyeorgos. At the end we prove that the corresponding two-parametric family of Riemannian metrics on SO(5)/U (2) = Sp(2)/U (1) · Sp(1) satisfying the above mentioned (strict!) inequalities, really generates δ-homogeneous spaces, which are not normal and are not naturally reductive with respect to any isometry group.2000 Mathematical Subject Classification: 53C20 (primary), 53C25, 53C35 (secondary).

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Section: Theorem 23 Implies Easily the Classification Of Transitive A...mentioning

confidence: 87%

On δ-homogeneous Riemannian manifolds

Berestovskiî¹,

Nikonorov²

2008

Differential Geometry and its Applications

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“…There is one further inequivalent (We take the volume to be V = y 2 1 y 2 y 2 3 .) These metrics were found in [15,16], and discussed further in [14]. The first metric is just the standard SO(7)-invariant metric on the Grassmanian SO(7)/[SO(2)×SO( 5)] [14].…”

Section: G 2 /Su(2) Maxmentioning

confidence: 93%

“…We find the Einstein constant and the invariant The metric on the coset G 2 /[SU (2) L ×U (1) R ] is obtained by dividing out the y 4 terms and the last of the three y 3 terms in (5.9): subgroups of G 2 ). The resulting coset space is isomorphic to the Grassmannian G + 2 (R 7 ) = SO(7)/[SO(2)×SO( 5)] of oriented 2-planes through the origin in R 7 [14].…”

Section: G 2 /Su(2) Diagmentioning

confidence: 99%

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Einstein metrics on group manifolds and cosets

Gibbons

Lü

Pope

2011

Journal of Geometry and Physics

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It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under G L × G R . Less well known is that every compact simple group manifold except SO(3) and SU (2) admits at least one more homogeneous Einstein metric, invariant still under G L but with some, or all, of the right-acting symmetry broken. (SO (3) and SU (2) are exceptional in admitting only the one, bi-invariant, Einstein metric.) In this paper, we look for Einstein metrics on three relatively low dimensional examples, namely G = SU (3), SO(5) and G 2 . For G = SU (3), we find just the two already known inequivalent Einstein metrics. For G = SO(5), we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G = G 2 we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G/H for the above groups G. In particular, for SO(5)/U (1) we find, depending on the embedding of the U (1), generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature (2, 6) on SU (3), an Einstein metric of signature (5, 6) on G 2 /SU (2) diag , and an Einstein metric of signature (4, 6) on G 2 /U (2). Interestingly, there are no Lorentzian Einstein metrics among our examples.

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“…in a pure-state controllable manner) on the homogeneous space SO(2d)/U(d) assuming d ≥ 3. The case d ≥ 4 is discussed in [60]. For d = 3 we have SO(6) ∼ = SU(4) and SU(4)/U(3) = CP 3 (where CP 3 denotes the complex projective space in four dimensions), and it is known that only subgroups of SU(4) isomorphic to SU(4) or Sp(2) ∼ = SO(5) can act transitively on CP 3 (see p. 168 of [61] or p. 68 of [62]; refer also to [63]).…”

Section: B Orbits and Stabilizers Of Quasifree States Under The Actimentioning

confidence: 99%

A dynamic systems approach to fermions and their relation to spins

Zimborás

Zeier

Keyl

et al. 2014

EPJ Quantum Technol.

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The key dynamic properties of fermionic systems, like controllability, reachability, and simulability, are investigated in a general Lie-theoretical frame for quantum systems theory. It just requires knowing drift and control Hamiltonians of an experimental set-up. Then one can easily determine all the states that can be reached from any given initial state. Likewise all the quantum operations that can be simulated with a given set-up can be identified. Observing the parity superselection rule, we treat the fully controllable and quasifree cases of fermions, as well as various translation-invariant and particle-number conserving cases. We determine the respective dynamic system Lie algebras to express reachable sets of pure (and mixed) states by explicit orbit manifolds.

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Some New Homogeneous Einstein Metrics on Symmetric Spaces

Cited by 28 publications

References 11 publications

On δ-homogeneous Riemannian manifolds

On δ-homogeneous Riemannian manifolds

Einstein metrics on group manifolds and cosets

A dynamic systems approach to fermions and their relation to spins

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