METRICS, CONNECTION S AN D GLUIN G THEOREM S 15 (A Baire se t i s a countabl e unio n o f open , dens e sets . I n particular , such a set i s dense. ) This theore m i s a direc t consequenc e o f (2.9.1) . T o explain , I mus t introduce th e se t o f equivalenc e classe s o f covarian t derivative s (a s in (2.10)) , wher e s i s constraine d t o b e unitar y (wit h determinan t 1 ) and als o to equal the identity a t p. Thi s set has a natural structur e o f a smooth, infinit e dimensiona l manifol d wit h a smoot h actio n o f 5/7(2) . In fact, (TB°|[v])/TSI7(2 ) i s the space of smooth sections of su(V)®T*X which obe y (2.11) d v *a = 0. (This las t equatio n introduce s th e * endomorphism o f A*T*X thi s endomorphism send s A P T*X t o A 4 ~PT*X an d i t i s defined b y th e rul e (2.12) a A * a = |a| 2 -dvol . Here, dvo l i s th e metric' s volum e form . (Thus , * 1 dvol. ) Not e tha t this * maps A 2 T*X t o itsel f wit h squar e 1 , and A ± ar e exactl y it s ± 1 eigenspaces.) By the way , th e literatur e i s consistent i n discussing certai n Sobole v space completions of above . Th e tangent spac e to such a completio n is define d a s i n (2.11 ) sav e tha t a i s an y Sobole v clas s L\ sectio n o f su(V) ® T*X fo r som e fixed k 1 . Tha t is , a and it s covarian t deriva - tives t o orde r k (usin g som e fixe d covarian t derivative ) ar e require d t o be square integrable over X. Th e introduction o f Sobolev spaces at thi s point prove d convenien t fo r variou s technica l reasons , bu t the y ar e no t in an y sens e fundamental t o th e presentation . To retur n t o .M 0 , observ e tha t A4° sit s i n an d s o inherit s th e subspace topology . Now , Ai° get s it s manifol d structur e fro m th e fac t that i t is cut ou t of as the zero set of a section, s , over of a certai n infinite dimensiona l vecto r bundle . This vector bundle, is the set o f equivalence classes of pairs (V,u ) where V i s a covarian t derivativ e o n V an d wher e u i s a sectio n o f su(V) g A+. Th e equivalenc e relatio n i s (2.13) (V,CJ ) - (s'V-s-^s-ws- 1 ) where s i s an y unitar y automorphis m o f V wit h determinan t 1 whic h equals I a t p . (Us e [V,CJ ] t o denot e th e equivalenc e clas s o f a pai r (V,u ).) Note , b y the way , that thi s vecto r bundl e ha s a natural SU(2) action whic h cover s the actio n o n B°.
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