Generalized Calabi-Yau manifolds(20)

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

thederivativeDX:U→EndSde nesanintegrablealmostcomplexstructureJonU

Proof:Sinceiφ(ρ)= , ¯ ,di erentiatingalongacurveinU,

˙= ,˙ .iφ˙ ¯ + , ¯

Uptoascalar,thepurespinorsformanorbit,soateachpoint

˙=c +σ(a)

forsomec∈Canda∈so(12,C).Butthen

,˙ =c , + σ(a) , =0(12)

wherethe rsttermiszerobecausethebilinearformisskewandthesecondbecause,aswesawabove,ing(12)

˙˙ = ,˙ =iφ. ,¯ ˙+ ¯˙ ¯ + , ¯

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