The geometry of compact homogeneous spaces with two isotropy summands

Dickinson, William; Kerr, Megan M.

doi:10.1007/s10455-008-9109-9

Cited by 59 publications

(87 citation statements)

References 27 publications

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“…In order to solve system (22), first, we need to determine the non zero triples c 2 11 , c 3 12 , c 4 13 , and c 4 22 of Proposition 8. By Theorem 2, the metric x 1 = 1, x 2 = 2, x 3 = 3, x 4 = 4 is Kähler-Einstein; thus, by using (20) and Table 5 we obtain the following: In order to compute c 4 22 = [224], we use the twistor fibration of generalized flag manifolds M = G/K over symmetric spaces G/U [18], [19, p. 48].…”

Section: Proposition 8 Let M = G/k Be a Generalized Flag Manifold Of mentioning

confidence: 99%

“…[10, p. 120]). As a consequence, the Einstein [8,20] G 2 /U (2)(U (2) represented by the short root) [8,20] [8,20] E 8 /SO (14) × U (1) s = 2 1 − =2 [8,20] SU [27,29] G 2 /U (2) (U (2) represented by the long root) [27] (1) SU (n)/T (n ≥ 4) s = n(n − 1)/2 ≥ n!/2 + 1 + E 1 [6] SU (2m)/T (m ≥ 2) s = m(2m − 1) ≥ (2m)!/2 + 1 + E 1 + E 2 [34] SU (2m)/T (m ≥ 6) s = m(2m − 1) ≥ (2m)!/2 + 1 + E 1 + E 2 + E 3 [21] SU (2m + 1)/T (m ≥ 6) s = m(2m + 1) ≥ (2m + 1)!/2 + 1 + E 1 + E 2 + E 3 SO(5)/T s = 4 -≥ 6 [34] SO(2n + 1)/T (n ≥ 12)…”

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confidence: 98%

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Invariant Einstein metrics on flag manifolds with four isotropy summands

2009

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Section: Proposition 8 Let M = G/k Be a Generalized Flag Manifold Of mentioning

confidence: 99%

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confidence: 98%

Invariant Einstein metrics on flag manifolds with four isotropy summands

2009

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“…A.24 Table 4 a Metrics also obtained by Dickinson and Kerr in [9] b idem for p = 1, 3. For p = 4, one of the metrics is the standard metric obtained by Wang and Ziller in [20] c The standard metric was obtained by Wang and Ziller in [20] (so 2n , so 2 p ⊕ so 2(n− p) , u p ⊕ so 2(n− p) ), (so 2n+1 , so 2 p+1 ⊕ so 2(n− p) , so 2 p+1 ⊕ so n− p ⊕ so n− p ), n − p even,…”

Section: Theorem 12mentioning

confidence: 99%

“…Due to this, most of the known examples of existence or non-existence of Einstein metrics are homogeneous spaces. For example, every isotropy irreducible space is clearly an Einstein manifold and recently Einstein metrics on homogeneous spaces with exactly two isotropy summands were classified by Dickinson and Kerr [9]. Einstein metrics on spheres and projective spaces were classified by Ziller [24] and Einstein normal homogeneous manifolds were classified by Wang and Ziller [20].…”

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confidence: 99%

Einstein homogeneous bisymmetric fibrations

Araujo

2011

Geom Dedicata

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We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K /L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.

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“…If there exists an intermediate subalgebra k, between h and g, then there exists at least a 2-parameter family of G-invariant metrics, and many spaces with exactly 2-parameters arise from such an intermediate subalgebra; such spaces were classified in [3]. Thus, Theorem 0.1 addresses the simplest nontrivial case of our classification problem.…”

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confidence: 99%

Homogeneous Metrics with Nonnegative Curvature

Schwachhöfer

Tapp

2009

J Geom Anal

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Abstract. Given compact Lie groups H ⊂ G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K, H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.Let H ⊂ G be compact Lie groups, with Lie algebras h ⊂ g, and let g 0 be a bi-invariant metric on G. The space G/H with the induced normal homogeneous metric, denoted (G, g 0 )/H, has nonnegative sectional curvature. Little is known about which other G-invariant metrics on G/H have nonnegative sectional curvature, except in certain cases. In all cases where G/H admits a G-invariant metric of positive curvature, the problem has been studied along with the determination of which G-invariant metric has the best pinching constant; see [9],[10], [11]. When H is trivial, this problem was solved for G = SO(3) and U (2) in [1], and partial results for G = SO(4) were obtained in [5]. Henceforth, we identify G-invariant metrics on G/H with Ad H -invariant inner products on p = the g 0 -orthogonal complement of h in g.In Section 1, it is an easy application of Cheeger's method to prove that the solution space is star-shaped. That is, if g is a G-invariant metric on G/H with nonnegative curvature, then the inverse-linear path, g(t), from g(0) = g 0 | p to g(1) = g is through nonnegatively curved Ginvariant metrics. Here, a path of inner products on p is called "inverse-linear" if the inverses of the associated path of symmetric matrices form a straight line. This observation reduces our problem to an infinitesimal one: first classify the directions, g ′ (0), one can move away from the normal homogeneous metric such that the inverse-linear path g(t) appears (up to derivative information at t = 0) to remain nonnegatively curved. Then, for each candidate direction, check how far nonnegative curvature is maintained along that path. In Section 2, we derive curvature variation formulas necessary to implement this strategy, inspired by power series derived by Müter for curvature along an inverse-linear path [8].In Section 3, we consider an intermediate subgroup K between H and G, with subalgebra k, so we have inclusions h ⊂ k ⊂ g. Write p = m ⊕ s, where m is the orthogonal compliment of h in k and s is the orthogonal compliment of k in g. The inverse-linear path of G-invariant * Supported by the Schwerpunktprogramm Differentialgeometrie of the Deutsche Forschungsgesellschaft.

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The geometry of compact homogeneous spaces with two isotropy summands

Cited by 59 publications

References 27 publications

Invariant Einstein metrics on flag manifolds with four isotropy summands

Invariant Einstein metrics on flag manifolds with four isotropy summands

Einstein homogeneous bisymmetric fibrations

Homogeneous Metrics with Nonnegative Curvature

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