A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternioni...

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Bibliographic Details
Published inAnnali di matematica pura ed applicata Vol. 184; no. 1; pp. 53 - 74
Main Authors Alekseevsky, D.V., Marchiafava, S.
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.03.2005
Subjects
Analytic functions
Construction
Manifolds (mathematics)
Mathematical analysis
Riemann manifold
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Summary:A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold of non zero scalar curvature, in particular, when is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M2n of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M2n of is the projection of a holomorphic Legendrian submanifold of the twistor space of , considered as a complex contact manifold with the natural holomorphic contact structure . Any Legendrian submanifold of the twistor space is defined by a generating holomorphic function. This is a natural generalization of Bryant’s construction of superminimal surfaces in S4=ℍP1.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-003-0089-x