Homogeneous Ricci positive 5-manifolds

Alekseevsky, Dmitri V.; Miatello, Isabel Dotti; Ferraris, Carlos J.

doi:10.2140/pjm.1996.175.1

Cited by 43 publications

(57 citation statements)

References 3 publications

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“…The metric g and orientation on N determine a metric and orientation on the 4-plane field H := ker η and hence the space of "horizontal" 2-forms Λ 2 H * splits as a direct sum of self-dual and anti-self-dual horizontal forms: Λ 2 H * = Λ + ⊕ Λ − . A triple (ω 1 , ω 2 , ω 3 ) satisfying (i) determines a trivialisation of Λ + and therefore a reduction of the structure group from SO(4) = SU (2) + · SU(2) − to SU(2) − . In fact, we can always assume that (ω 1 , ω 2 , ω 3 ) is an oriented basis of Λ + with respect to the natural orientation induced from the orientation of ker η; this gives condition (ii) above.…”

Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

“…Underlying this is the low dimensional isomorphism Spin(5) ∼ = Sp(2) and the fact that the spinor representation of Spin(5) is isomorphic to the fundamental representation of Sp(2) on H 2 . Hence the isotropy subgroup of a nonzero spinor ψ in five dimensions is isomorphic to Sp(1) ∼ = SU (2). It follows that an SU(2)-structure on a 5-manifold N is equivalent to the choice of a spin structure on N and a unit spinor.…”

Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

“…Consider now a (not necessarily complete or compact) 6-manifold M with an SU(3)-structure preserved by a cohomogeneity one isometric action of a compact Lie group G. For reasons explained at the beginning of Section 3, we will assume G = SU(2) × SU (2) and that a dense open set M * ⊂ M is diffeomorphic to the product of an interval with the 5-dimensional homogeneous space…”

Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

“…The Calabi-Yau structures on the conifold itself and on both of its desingularisations are cohomogeneity one. The group acting is SU(2)×SU (2) and the generic (principal) orbit is diffeomorphic to SU(2) × SU(2)/ U(1) S 2 × S 3 in all three cases. The singularity of the conifold is replaced by a round totally geodesic holomorphic S 2 in the small resolution and by a round totally geodesic special Lagrangian S 3 in the smoothing.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we parametrise the space of nearly hypo structures on S 2 × S 3 invariant under SU(2) × SU (2), showing that it is a smooth connected 4-manifold. In Section 3 we derive the ODE system (3.10) whose solutions are cohomogeneity one nearly Kähler structures.…”

Section: Introductionmentioning

confidence: 99%

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New $\mathrm{G}_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

2017

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Abstract. There is a rich theory of so-called (strict) nearly Kähler manifolds, almostHermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kähler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6-manifold.A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions. IntroductionAt least since the early 1950s (see Steenrod's 1951 book [45, 41.22]) it has been well known that viewing S 6 as the unit sphere in Im O endows it with a natural nonintegrable almost complex structure J defined via octonionic multiplication. Since J is compatible with the round metric g rd , the triple (g rd , J, ω), where ω(·, ·) = g rd (J·, ·), defines an almost-Hermitian structure on S 6 . Its torsion has very special properties: in particular, dω is the real part of a complex volume form Ω. Appropriately normalised, the pair (ω, Ω) defines an SU(3)-structure on S 6 which by construction is invariant under the exceptional compact Lie group G 2 Aut(O).Octonionic multiplication also defines a G 2 -invariant 3-form ϕ on Im O byWe call this the standard G 2 -structure on R 7 . Regarding R 7 as the Riemannian cone over (S 6 , g rd ), ϕ and its Hodge dual are given in terms of (ω, Ω):Conversely, viewing S 6 as the level set r = 1 in R 7 , the SU(3)-structure (ω, Ω) is recovered from ϕ and * ϕ by restriction and contraction by the scaling vector field ∂ ∂r . More generally, consider a 7-dimensional Riemannian cone C = C(M ) over a smooth compact manifold (M 6 , g). Suppose that the holonomy of the cone is contained in G 2 . Then there exists a pair of closed (in fact parallel) differential forms ϕ and * ϕ, pointwise equivalent to the model forms on R 7 and homogeneous with respect to scalings on C. Just as above, viewing M as the level set r = 1 in C, the restriction and contraction by ∂ ∂r of ϕ and * ϕ define an SU(3)-structure (ω, Ω) on M satisfying (1.1). In particular, the closedness of ϕ and * ϕ is equivalent to There are other possible equivalent definitions of a nearly Kähler 6-manifold. By relating the holonomy reduction of the cone C(M ) to the existence of a parallel spinor instead of a pair of distinguished parallel forms, nearly Kähler 6-manifolds can be characterised as those 6-manifolds admitting a real Killing spinor [34]. Alternatively, one could give a definition in terms of Gray-Hervella torsion classes of almost Her...

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Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

Section: Cohomogeneity One Su(3)-structures and Homogeneous Su(2)-strmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New $\mathrm{G}_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

2017

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All homogeneous N = 2 M-theory truncations with supersymmetric AdS4 vacua

Cassani

Koerber

Varela

2012

J. High Energ. Phys.

146

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We study consistent truncations of M-theory to gauged N = 2 supergravity in four dimensions, based on a large class of SU(3)-structures in seven dimensions. We show that the gauging involves isometries of the vector multiplet scalar manifold as well as the Heisenberg algebra and a special isometry of the hyperscalar manifold. As a result, non-abelian gauge groups and new non-trivial scalar potentials are generated. Then we specialize to all homogeneous SU(3)-structures supporting supersymmetric AdS 4 vacua. These are the Stiefel manifold V 5,2 , the Aloff-Wallach spaces N(k, l), the seven-sphere (seen as SU(4)/SU(3) or Sp(2)/Sp(1)) and the M 110 and Q 111 coset spaces. For each of these cases, we describe in detail the N = 2 model and discuss its peculiarities.

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New Invariant Einstein–Randers Metrics on Stiefel Manifolds$$V_{2p}\mathbb {R}^n={\mathrm {S}}{\mathrm O}(n)/ {\mathrm {S}}{\mathrm O} (n-2p)$$

Chen

2021

Results Math

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Homogeneous Ricci positive 5-manifolds

Abstract: We classify all 5-dimensional homogeneous Riemannian manifolds with positive Ricci curvature and among these we determine all Einstein manifolds. A new Einstein metric is found.

Cited by 43 publications

References 3 publications

New $\mathrm{G}_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

New $\mathrm{G}_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

All homogeneous N = 2 M-theory truncations with supersymmetric AdS4 vacua

New Invariant Einstein–Randers Metrics on Stiefel Manifolds$$V_{2p}\mathbb {R}^n={\mathrm {S}}{\mathrm O}(n)/ {\mathrm {S}}{\mathrm O} (n-2p)$$

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