Unit Killing Vector Fields on Nearly Kähler Manifolds

Moroianu, Andrei; Nagy, Paul-Andi; Semmelmann, Uwe

doi:10.1142/s0129167x05002874

Cited by 33 publications

(31 citation statements)

References 15 publications

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“…A similar situation has been studied in [1], in the context of seven-dimensional manifolds with holonomy G 2 . A special case has been studied in [12], where it was proved that there is only one complete example of a nearlyKähler SU(3)-structure with a Killing vector field of unit norm, namely the standard nearly-Kähler structure on S 3 × S 3 . The paper is divided into two parts.…”

Section: Introductionmentioning

confidence: 99%

Special Symplectic Six-Manifolds

Conti

Томассини

2007

The Quarterly Journal of Mathematics

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Section: Introductionmentioning

confidence: 99%

Special Symplectic Six-Manifolds

Conti

Томассини

2007

The Quarterly Journal of Mathematics

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“…This determines a parallel tractor metric H on M via the formula (59). On M the density field τ := H AB X A X B is nowhere zero and H, τ and the Einstein metric are related as in the expression (56). Now p| M is the restriction of a smooth projective structure p on M .…”

Section: 5mentioning

confidence: 99%

Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

Gover¹,

Panai²,

Willse³

2017

Indiana Univ. Math. J.

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Abstract. We show that any dimension 6 nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a G ( * ) 2 holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel 7-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly Kähler manifold, if the cross product is definite; otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly Kähler and nearly para-Kähler parts separated by a hypersurface which canonically carries a Cartan (2, 3, 5)-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para-)Kähler structures. Any real-analytic (2, 3, 5)-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure.A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry; this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para-)Kähler manifolds, including how the almost (para-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.

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“…Throughout the following we will always suppose that G preserves the almost complex structure J . Note that this condition is automatically satisfied if the manifold is compact and the metric has not constant sectional curvature (see [22], Prop. 3.1).…”

Section: Theorem 22mentioning

confidence: 99%

“…A NK structure (g, J ) is called strict if ∇ v J | p = 0 for every p ∈ M and every 0 = v ∈ T p M (see e.g. [13,14,[22][23][24] for main properties). Significant examples are the so-called 3-symmetric spaces with their canonical almost complex structures.…”

Section: Introductionmentioning

confidence: 99%

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

Podestà

Spiro

2012

Commun. Math. Phys.

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Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G = SU 2 ×SU 2 , and M reg ⊂ M its subset of regular points. We show that M reg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kähler structures and that a 1-parameter subfamily of such structures smoothly extends over a singular orbit of type S 3 . This determines a new class of examples of nearly Kähler structures on T S 3 .

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Unit Killing Vector Fields on Nearly Kähler Manifolds

Abstract: Abstract. We study 6-dimensional nearly Kähler manifolds admitting a Killing vector field of unit length. In the compact case it is shown that up to a finite cover there is only one geometry possible, that of the 3-symmetric space S 3 × S 3 .

Cited by 33 publications

References 15 publications

Special Symplectic Six-Manifolds

Special Symplectic Six-Manifolds

Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

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