Chern-flat and Ricci-flat invariant almost Hermitian structures

Scala, Antonio J. Di; Vezzoni, Luigi

doi:10.1007/s10455-010-9243-z

Cited by 18 publications

(15 citation statements)

References 23 publications

(25 reference statements)

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“…Proof. Let ∇ be the Levi-Civita connection of g. From ∇ 1 ≡ 0 and equation (15), it follows that ∇ x J = −J∇ x for every x ∈ g, therefore (16) g (∇ x Jy, z) = −g (J∇ x y, z) = g (∇ x y, Jz) ,…”

Section: 2mentioning

confidence: 99%

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Abelian Hermitian geometry

Andrada

Barberis

Dotti

2012

Differential Geometry and its Applications

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We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.2010 Mathematics Subject Classification. 53C15, 53B35, 53C30.

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Section: 2mentioning

confidence: 99%

“…A similar result for the Chern connection does not hold. See [15] for results concerning the flatness of the Chern connection on nilmanifolds.…”

Section: 2mentioning

confidence: 99%

Abelian Hermitian geometry

Andrada

Barberis

Dotti

2012

Differential Geometry and its Applications

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“…Furthermore, I'm grateful to Gueo Grantcharov for useful conversations and remarks and to Nicola Enrietti for an important observation on the presentation of the main result. [20] and Proposition 4.10 and 4.11 of [10]. Moreover things work differently either in the 3-step nilpotent case or in the 2-step solvable case (see [10]).…”

mentioning

confidence: 99%

“…[20] and Proposition 4.10 and 4.11 of [10]. Moreover things work differently either in the 3-step nilpotent case or in the 2-step solvable case (see [10]).…”

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confidence: 99%

A note on canonical Ricci forms on $2$-step nilmanifolds

Vezzoni

2012

Proc. Amer. Math. Soc.

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In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci -flat with respect to another canonical connection if and only if it is cosymplectic (i.e. d * ω = 0). introductionLet (M, g, J, ω) be an almost Hermitian manifold. Gauduchon introduced in [13] a 1-parameter family ∇ t of canonical Hermitian connections which can be distinguished by the properties of the torsion tensor T . In this family ∇ 1 corresponds to so-called Chern connection which can be defined as the unique Hermitian connection whose (1, 1)-part of the torsion vanishes. In the quasi-Kähler case (i.e. when ∂ω = 0), the line {∇ t } degenerates to a single point and the Chern connection is the unique canonical connection.Any canonical connection ∇ t induces the so-called Ricci form ρ t (X, Y ) = 2 i tr ω R t (X, Y ), where R t denotes the curvature of ∇ t . It turns out that ρ t is always a closed form which can be locally written as the derivative of the 1-form θ t (X) = n r=1 g(∇ t X Z r , Z r ), where {Z r } is a (local) unitary frame. Moreover, in the cosymplectic case (i.e. when dω n−1 = 0) the line {θ t } degenerates to a single point (see Corollary 3.3) and all the canonical connections have the same Ricci form.The aim of this paper is to study the Ricci forms ρ t on 2-step nilmanifolds equipped with a left-invariant almost Hermitian structure. We recall that by definition a k-step nilmanifold is a compact quotient of a k-step nilpotent Lie group G by lattice. Since we are considering left-invariant almost Hermitian structures, we can work on Lie algrebras in an algebraic fashion. Our main result is the following Theorem 1. Let (g, g, J, ω) be a 2-step nilpotent Lie algebra with an almost Hermitian structure. Then (g, J) is Ricci-flat with respect to the Chern connection and it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic (i.e. d * ω = 0). This theorem has the following immediate consequence: Corollary 1.1. Every left-invariant almost Hermitian structure on a nilmanifold associated to a 2-step Lie group is Ricci-flat with respect to the Chern connection.

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“…In terms of the Levi Civita connection D of g , the Chern connection is given by

g (\nabla_{X} Y, Z) = g (D_{X} Y, Z) - \frac{1}{2} d ω (J X, Y, Z) .

We refer to e.g. [, (2.1)], [, (2.1)] and [, Section ] for different equivalent descriptions. Note that

\nabla = D

if and only if

(M, J, ω, g)

is Kähler.…”

Section: Chern‐ricci Formmentioning

confidence: 99%

On the Chern‐Ricci flow and its solitons for Lie groups

Lauret

Valencia

2015

Mathematische Nachrichten

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Abstract. This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern-Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.

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Chern-flat and Ricci-flat invariant almost Hermitian structures

Abstract: We study left-invariant almost Hermitian structures on homogeneous spaces having either flat Chern connection or flat Ricci-Chern form. Many examples are carefully described, and a classification is given in low dimensions.

Cited by 18 publications

References 23 publications

Abelian Hermitian geometry

Abelian Hermitian geometry

A note on canonical Ricci forms on $2$-step nilmanifolds

On the Chern‐Ricci flow and its solitons for Lie groups

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