Self-dual Einstein metrics

Vitter, Al

doi:10.1090/conm/051/848939

Cited by 7 publications

(2 citation statements)

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“…Thus, if M is compact, the twistor space (Z, h t , J 2 ) gives a negative answer to the Blair-Ianuş question. Examples of compact Einstein self-dual manifolds with negative scalar curvature can be found in [97]. Multiplying the twistor space of such a manifold by Kähler manifolds, one can construct examples of non-Kähler almost Kähler manifolds of arbitrary even dimension ≥ 6 which have Hermitian Ricci tensors.…”

Section: Twistor Spaces With Hermitian Ricci Tensormentioning

confidence: 99%

“…Proposition 3, which gives twistorial examples of such manifolds, seems to be interesting in the case of negative scalar curvature of (M, g) since the complete classification of compact Einstein self-dual manifolds with negative scalar curvature is not available yet. It has been conjectured by A. Vitter [97] that every such a manifold is a quotient of the unit ball in C 2 with the metric of negative constant sectional curvature or the Bergman metric. Corollary 3 can be used to show that an isometry of the twistor space preserves vertical, and hence horizontal, spaces.…”

Section: Corollary 2 ([36]mentioning

confidence: 99%

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Curvature Properties of the Chern Connection of Twistor Spaces

Davidov¹,

Grantcharov²,

Mus˘karov³

2009

Rocky Mountain J. Math.

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The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h t compatible with the almost complex structures J 1 and J 2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z, h t , J n ), n = 1, 2 and find the geometric conditions on M under which the curvature of its Chern connection D n is of type (1, 1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D n and show that the Nijenhuis tensor of J 2 is D 2parallel provided the base manifold M is Einstein and self-dual.

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Section: Twistor Spaces With Hermitian Ricci Tensormentioning

confidence: 99%

Section: Corollary 2 ([36]mentioning

confidence: 99%