Hyper-Kähler quotients of solvable Lie groups

Barberis, M. L.; Dotti, Isabel G.; Fino, Anna

doi:10.1016/j.geomphys.2005.04.013

Cited by 26 publications

(40 citation statements)

References 17 publications

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“…In this section we recall some results on flat left-invariant Lie groups due to Milnor [14] and Barberis-Dotti-Fino [3]. These results lead to Corollary 1.6 stated in Section 1.…”

Section: Appendix a Lie Groups With A Flat Left-invariant Kähler Strmentioning

confidence: 77%

“…In fact, there are examples of non-abelian Lie groups which admits left invariant flat Kähler structures, see e.g. [3] and the appendix at the end of the present paper. The results in [3] give a Kähler analogue of Milnor's classification theorem on flat Lie groups with left-invariant Riemannian metrics [14].…”

mentioning

confidence: 89%

“…In the above theorem, the abelian ideal i is i = {Y ∈ g | ∇ Y = 0}, where ∇ is the Levi-Civita connection. It is further observed (Proposition 2.1 in [3]) that i = c ⊕ [g, g] where c is the center of g and [g, g] is of even dimension.…”

Section: Appendix a Lie Groups With A Flat Left-invariant Kähler Strmentioning

confidence: 99%

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Lie groups with flat Gauduchon connections

2019

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We pursuit the research line proposed in [24] about the classification of Hermitian manifolds whose s-Gauduchon connection ∇ s = (1 − s 2 )∇ c + s 2 ∇ b is flat, where s ∈ Ê and ∇ c and ∇ b are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected 2n-dimensional Lie group G equipped with a left-invariant complex structure J and a leftinvariant compatible metric g and we assume that its connection ∇ s is flat. Our main result states that if either n=2 or there exits a ∇ s -parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with ∇ s -flat Hermitian metrics.In the following given a real number s, we call ∇ s as the s-Gauduchon connection. Note that ∇ 0 = ∇ c and ∇ 2 = ∇ b . Moreover, ∇ 1 corresponds to the projection of the Levi-Civita connection onto the space of (1, 0)-vector fields. This connection is called the first canonical Hermitian connection in [8] and [13], the associated connection in [9] and the complexified Levi-Civita connection in [12]. In [24], it is proposed the study of compact Hermitian manifolds with flat s-Gauduchon connections, as a special case of a problem posed by Yau (Problem 87 in [26]). The case s = 0 (i.e. when the Chern connection is flat) was studied in the 1950s by W. Boothby [5] and H. C. Wang [20]. Boothby proved that if (M n , g) is a compact Chern-flat 2010 Mathematics Subject Classification. 53C55 (Primary).

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“…In this section we recall some results on flat left-invariant Lie groups due to Milnor [14] and Barberis-Dotti-Fino [3]. These results lead to Corollary 1.6 stated in Section 1.…”

Section: Appendix a Lie Groups With A Flat Left-invariant Kähler Strmentioning

confidence: 77%

mentioning

confidence: 89%

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Lie groups with flat Gauduchon connections

2019

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“…Compare Section 7.2.2. For a discussion of the structure of flat Kähler Lie groups, see for example [14] or [3]. Remark 1.3.…”

Section: 4 Theorem 43)mentioning

confidence: 99%

“…The proof of Proposition 7.6 shows that the classification of groups N (k, l) up to isomorphism of Sasaki Lie groups amounts exactly to the classification of Kähler Lie groups C(k, l) up to isomorphism. For a discussion of the structure of flat Kähler Lie groups, see for example [3] and [14].…”

Section: Heisenberg Modificationsmentioning

confidence: 99%

Locally homogeneous aspherical Sasaki manifolds

Baues

Kamishima

2020

Differential Geometry and its Applications

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New HKT manifolds arising from quaternionic representations

Barberis

Fino

2009

Math. Z.

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Abstract. We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-Kähler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in SL(n, H) which is not a nilmanifold. We find in addition new compact strong HKT manifolds.We also show that every Kähler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional Kähler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8.Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.

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Hyper-Kähler quotients of solvable Lie groups

Cited by 26 publications

References 17 publications

Lie groups with flat Gauduchon connections

Lie groups with flat Gauduchon connections

Locally homogeneous aspherical Sasaki manifolds

New HKT manifolds arising from quaternionic representations

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