Abelian Hermitian geometry

Andrada, Adrián; Barberis, M. L.; Dotti, Isabel G.

doi:10.1016/j.difgeo.2012.07.001

Cited by 22 publications

(27 citation statements)

References 19 publications

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“…• They generalize previous results concerning complex product structures [2,7], complex and symplectic structures related to tangent algebras [1,6,17], complex and paracomplex structures on homogeneous manifolds [11]. • The existence of LSA structures imposes a clear obstruction.…”

Section: Introductionsupporting

confidence: 79%

From almost (para)-complex structures to affine structures on Lie groups

Calvaruso

Ovando

2017

manuscripta math.

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Let G = H ⋉ K denote a semidirect product Lie group with Lie algebra g = h⊕k, where k is an ideal and h is a subalgebra of the same dimension as k. There exist some natural split isomorphisms S with S 2 = ± Id on g: given any linear isomorphism j : h → k, we get the almost complex structure J(x, v) = (−j −1 v, jx) and the almost paracomplex structure E(x, v) = (j −1 v, jx). In this work we show that the integrability of the structures J and E above is equivalent to the existence of a left-invariant torsionfree connection ∇ on G such that ∇J = 0 = ∇E and also to the existence of an affine structure on H. Applications include complex, paracomplex and symplectic geometries.2010 Mathematics Subject Classification. 53C15, 53C55, 53D05, 22E25, 17B56.

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Section: Introductionsupporting

confidence: 79%

From almost (para)-complex structures to affine structures on Lie groups

Calvaruso

Ovando

2017

manuscripta math.

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“…Also ∇ 1 -flat manifolds have been studied in several papers, and, even though there is no exhaustive classification of such manifolds, there are some partial classification results (see [1] and [9]).…”

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confidence: 99%

“…In particular, if a left invariant connection admits a left invariant parallel frame, then its torsion is parallel. By the result of Kamber and Tondeur [10] (see also [1]), given any smooth manifold M equipped with a complete affine connection ∇ which is flat and whose torsion is parallel, then the universal cover of M is a Lie group and ∇ lifts to a left invariant connection. Hence theorem 1.3 can be restated as follows: Theorem 1.4.…”

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confidence: 99%

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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“…Some properties of this kind of complex structures are stated in the following result (see [3,16] for their proofs).…”

Section: Special Casesmentioning

confidence: 99%

Locally conformally Kähler solvmanifolds: a survey

Andrada

Origlia

2019

Complex Manifolds

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A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.2010 Mathematics Subject Classification. 53B35, 53A30, 22E25.

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Abelian Hermitian geometry

Cited by 22 publications

References 19 publications

From almost (para)-complex structures to affine structures on Lie groups

From almost (para)-complex structures to affine structures on Lie groups

Lie groups with flat Gauduchon connections

Locally conformally Kähler solvmanifolds: a survey

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