Four-dimensional Kähler metrics admitting c-projective vector fields

Bolsinov, Alexey V.; Matveev, Vladimir S.; Mettler, Thomas; Rosemann, Steffan

doi:10.1016/j.matpur.2014.07.005

Cited by 11 publications

(11 citation statements)

References 19 publications

(73 reference statements)

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“…This Kähler structure is a special case of those obtained in [3,9]. It is straight-forward to check that g is Ricci-flat but non-flat and that the (1, 1)-tensor…”

Section: Proof Of Theorems 2 Andmentioning

confidence: 69%

Conification construction for Kähler manifolds and its application in c-projective geometry

Matveev

Rosemann

2015

Advances in Mathematics

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Abstract. Two Kähler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kähler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected 2n-dimensional Riemannian Kähler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we describe all possible dimensions of the space of essential c-projective vector fields of Kähler and Kähler-Einstein Riemannian metrics. We also show that two c-projectively equivalent Kähler Einstein metrics (of arbitrary signature) on a closed manifold have constant holomorphic curvature or are affinely equivalent.

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“…This Kähler structure is a special case of those obtained in [3,9]. It is straight-forward to check that g is Ricci-flat but non-flat and that the (1, 1)-tensor…”

Section: Proof Of Theorems 2 Andmentioning

confidence: 69%

Conification construction for Kähler manifolds and its application in c-projective geometry

Matveev

Rosemann

2015

Advances in Mathematics

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“…The local normal forms for compatible pairs (h, L) have been obtained in [9] and this combined with Theorem 1.5 implies the local normal forms for a c-compatible pair (g, ω) and A, see Example 5 and Theorem 1.6 below. We also refer to [10,Theorem 3.1] for the formulas in the 4-dimensional case.…”

Section: 3mentioning

confidence: 99%

“…For positive definite metrics, Theorem 1.1 was proved in [31], where also a history including a list of previously proven special cases can be found. The 4-dimensional version of Theorem 1.1 was recently proved in [10].…”

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confidence: 97%

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Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics

Bolsinov¹,

Matveev²,

Rosemann³

2021

asens

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Two Kähler metrics on a complex manifold are called c-projectively equivalent if their J-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected Kähler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is CP n with (a multiple of) the Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz conjecture for metrics of Lorentzian signature: we show that on a closed connected Lorentzian manifold, any projective vector field is an affine vector field. Contents 1. Introduction 1.1. Definitions and description of results 1.2. Main idea 1.3. Local description of c-projectively equivalent metrics 1.4. Structure of the paper 2. Canonical Killing vector fields for c-projectively equivalent metrics 3. Reduction to the real projective setting 3.1. The Kähler quotient w.r.t. a local isometric hamiltonian R -action 3.2. Reduction from the c-projective to the projective setting 4. Local description of c-projectively equivalent metrics 4.1. Local description of the quotients of c-projectively equivalent metrics and lifting 4.2. Realisation 4.3. Explicit formulas 5. Proof of the Yano-Obata conjecture (Theorem 1.1) 5.1. Conventions and degree of mobility 5.2. Scheme of the proof 5.3. C-projectively invariant form of equation ( 1.4) and special features of D(g, J) = 2 5.4. Properties of the eigenvalues of A 5.5. Explicit formulas for the non-constant block 5.6. There is only one non-constant eigenvalue 5.7. Proof of Theorem 1.1 when there is only one non-constant eigenvalue 6. Final step of the proof of Theorem 1.2: the case of a non-constant Jordan block Appendix A. Eigenvalues of the curvature operator References

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“…Independently of this and with different motivation, the theory of hamiltonian 2forms was developed locally and globally in [1,2]. In the more recent articles [6,7,16,17] ideas of both theories have been combined.…”

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confidence: 99%

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

Bolsinov

Rosemann

2021

Communications in Analysis and Geometry

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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.

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Four-dimensional Kähler metrics admitting c-projective vector fields

Cited by 11 publications

References 19 publications

Conification construction for Kähler manifolds and its application in c-projective geometry

Conification construction for Kähler manifolds and its application in c-projective geometry

Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics

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

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