The Bochner-flat cone of a CR manifold

David, Liana

doi:10.1112/s0010437x07003363

Cited by 3 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A case by case analysis shows that the only Abelian subgroup where the positivity condition can be satisfied is in that of a maximal torus. (This can be ascertained, for instance, by looking at Theorem 6 of [15]. )…”

Section: The Sasaki Conementioning

confidence: 99%

Kähler–Einstein Metrics

Boyer¹,

Galicki²

2007

Sasakian Geometry

View full text Add to dashboard Cite

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2 -norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that just the standard polarization can be represented by a Sasaki-Einstein metric.

show abstract

Section: The Sasaki Conementioning

confidence: 99%

Kähler–Einstein Metrics

Boyer¹,

Galicki²

2007

Sasakian Geometry

View full text Add to dashboard Cite

show abstract

“…the flow of a CR-vector field which is transversal to the Levi distribution. This observation is originally due to [38] (see also [18,19,34]) and this point of view was also taken up in [9] from a more general perspective to prove that also compact Bochner-flat (pseudo-)Kähler manifolds are locally symmetric. Theorem 1.2 provides a local description of Bochner-flat (pseudo-)Kähler manifolds on neighbourhoods of generic points which are characterized by some regularity condition, cf.…”

mentioning

confidence: 91%

Local description of Bochner-flat (pseudo-)Kähler metrics

Bolsinov

Rosemann

2021

Communications in Analysis and Geometry

View full text Add to dashboard Cite

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a by-product, we also describe all Kähler-Einstein metrics admitting a c-projectively equivalent one.

show abstract

“…It has constant Φ-sectional curvature equal to −3, where Φ is the endomorphism defining the natural CR structure on H 2n+1 . The relationship with the Heisenberg group has been noted in [BGM06,BGO07,BG08], and its appearance in CR spherical geometry was studied further in [Kam06,Dav08]. Indeed, Kamishima has noted the important connection between the CR structure on H 2n+1 and the Bochner-flat structures on C n classified by Bryant [Bry01].…”

Section: Introductionmentioning

confidence: 98%

The Sasakian Geometry of the Heisenberg Group

Boyer

2009

Preprint

View full text Add to dashboard Cite

In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has Φsectional curvature −3. I also briefly discuss some relations with the well-know sub-Riemannian geometry of the Heisenberg group as well as the standard Sasakian structure induced on compact quotients. Dedicated to Professor S. Ianus on the occasion of his 70th Birthday

show abstract

The Bochner-flat cone of a CR manifold

Cited by 3 publications

References 10 publications

Kähler–Einstein Metrics

Kähler–Einstein Metrics

Local description of Bochner-flat (pseudo-)Kähler metrics

The Sasakian Geometry of the Heisenberg Group

Contact Info

Product

Resources

About