Generalized Calabi-Yau manifolds

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

arXiv:math/0209099v1 [math.DG] 10 Sep 2002GeneralizedCalabi-YaumanifoldsNigelHitchinMathematicalInstitute24-29StGilesOxfordOX13LBUKhitchin@maths.ox.ac.ukFebruary1,2008AbstractAgeometricalstructureoneven-dimensionalmanifoldsisde nedwhichgener-alizesthenotionofaCalabi-Yaumanifoldandalsoasymplecticmanifold.Suchstructuresareofeitheroddoreventypeandcanbetransformedbytheactionofbothdi eomorphismsandclosed2-forms.Inthespecialcaseofsixdimen-sionswecharacterizethemascriticalpointsofanaturalvariationalproblemonclosedforms,andprovethatalocalmodulispaceisprovidedbyanopensetineithertheoddorevencohomology.1Introduction

WeintroduceinthispaperageometricalstructureonamanifoldwhichgeneralizesboththeconceptofaCalabi-Yaumanifold–acomplexmanifoldwithtrivialcanonicalbundle–andthatofasymplecticmanifold.Thisispossiblyausefulsettingforthebackgroundgeometryofrecentdevelopmentsinstringtheory,butthiswasnottheoriginalmotivationfortheauthor’s rstencounterwiththisstructure:itaroseinsteadaspartofaprogramme(followingthepapersforcharacterizingspecialgeometryinlowdimensionsbymeansofinvariantfunctionalsofdi erentialforms.Inthisrespect,thedimensionsixisparticularlyimportant.Thispaperhastwo

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