METRICS, CONNECTION S AN D GLUIN G THEOREM S 5 simplest exampl e whic h i s no t conformall y flat i s CP 2 wit h it s Fubini - Study metric . Here , CP 2 i s the quotien t o f the roun d 5-spher e i n C 3 (~ M6) b y th e S 1 actio n whic h simultaneousl y rotate s th e thre e comple x coordinates. Thus , X £ S 1 (a s the unit spher e in C) and (z 1? z2, z$) C 3 are sen t t o (Azi , Az2, Az3). Th e orientatio n her e i s the opposit e o f th e natural orientation which is induced on S 5 an d on S 1 a s the unit sphere s in complex vecto r spaces . The Fubini-Stud y metric , grs, i s obtaine d fro m th e roun d metri c (ground) on S 5 a s follows: Le t [z] b e a point i n CP 2 an d le t v b e a vecto r at [z\. Le t z b e a poin t i n S 5 whic h map s t o [z\. The n v ha s a uniqu e lift t o a vector v ' a t z whic h i s annihilated b y the 1-for m (1.7) G = 2- 1 e~ 1 ^2 E t i fe dzi - * d*). With thi s understood , defin e th e inne r produc t betwee n vector s v\ an d v2 a t [z] b y the rul e g F s{vuv2) = SroundK,^) . There ar e a fe w othe r examples , an d thes e ar e discusse d i n a subse - quent lecture . However, i f one changes th e questio n slightly , then ther e i s a genera l "existence" theore m fo r suc h metrics . Thi s theore m an d som e o f th e surrounding issue s is , in a sense, the subjec t matte r o f these lectures . Theorem 1 . ([44]) : Let M be a compact, oriented, dimension 4 man- ifold. Then there exists a positive integer N such that M# n CP 2 has metrics with W+ = 0 for all n N. For the uninitiated, th e connect su m XjfY betwee n manifolds X an d Y (o f the sam e dimension) i s defined a s follows: Tak e the interio r o f a n embedded close d bal l ou t o f X an d als o on e ou t o f Y. Th e resul t i s a pair o f manifold s wit h boundary , X' an d Y'. Th e boundarie s o f bot h are sphere s o f dimensio n on e les s tha n tha t o f X (o r Y) . Glu e X' an d Y' togethe r b y identifyin g thei r boundaries . When dim(-X') = dim(y ) = 4 , use the antipodal map to identify thes e boundary 3-spheres . Th e resul t wil l have, quite naturally, th e structur e of a smoot h manifol d whic h i s oriente d i f bot h X an d Y ar e oriented . (The us e of the antipoda l ma p insure s tha t th e inclusion s o f X' an d Y' into X # y i s orientation preserving. ) Her e i s a picture :
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