O C . TAUBE S up fo r th e discussio n o f Theorem 1 . f) A n exampl e Given belo w i s th e simples t exampl e o f a covarian t derivativ e wit h anti-self dua l connection . Tak e the bas e space X t o b e R 4 an d tak e th e vector bundl e i n question t o b e V = R 4 x C 2 . I f one thinks o f a sectio n of V a s a C 2 value d functio n o n R 4 , the n th e exterio r derivativ e o f such a functio n define s a covarian t derivative , d , o n ^ . Tha t is , ds (1.10) d s = E 4 = 1 ® d x \ OXi This covarian t derivativ e ha s R d = 0 . A thre e par t digressio n i s require d i n orde r t o presen t a les s trivia l example. Par t 1 of the digressio n introduce s th e Paul i matrice s f1'11* T l S ( l ~o) T 2 = G o) ' T 3 = ( o - i and se t r 4 t o denote the identity 2 x 2 matrix . Le t r\ denot e the adjoin t matrix. Par t 2 introduce s th e linea r functio n x : R 4 End(C 2 ) b y the rul e (1.12) X = E 4 j=i XiTi. Note tha t x 2 ~ —\x\ 2 r4. Us e the notatio n d x t o denot e E^x^r^ . Also , use xl t o denote the adjoint matrix , an d use dx1 t o denote S^da^r/. Par t 3 of the digressio n introduce s a final piec e of notation: Defin e (1.13) if i ^ 4 if i = 4 . Define a covarian t derivative , V , o n V usin g th e rul e « *\ r- r im(xdx* ) (1.14) V 8 s d 8 + J ^ . . 8 . The curvatur e o f this connectio n i s v _ im(dx A dx 1 ) ( L 1 5 j K = ( l + M 2 )2 ' which i s anti-self dual . (To put (1.14 ) int o perspective, not e tha t th e difference , a = V V', of any pair, V an d V' , o f covariant derivative s on some vector bundl e V over a manifold X i s automatically a section ove r X o f End(V) ® T*X. Conversely, i f V i s a covarian t derivativ e o n V an d i f a i s an y sectio n
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