METRICS, CONNECTION S AN D GLUIN G THEOREM S 9 of End(V ) (g T*X, the n on e ca n defin e a ne w covarian t derivativ e b y V = V + a. Thus , th e se t o f covarian t derivative s o n a fixed vecto r bundle over a fixed manifold X i s naturally a n affine spac e whose linear model i s the spac e o f sections of End(V) ® T*X.) The covarian t derivativ e i n (1.14 ) happen s t o b e metri c compatibl e with respec t t o th e simples t possibl e metri c o n V, namel y tha t whic h defines \v\ 2 t o equa l v t v. Furthermore , th e curvatur e i n (1.15 ) ha s van - ishing trace . The exampl e i n (1.14 ) i s known a s the " 1 instanton " solutio n t o th e anti-self dua l equations . I n fact , ther e i s a n interestin g 5 - paramete r family o f " 1-instanton" solutions . Th e parameter s ar e (A , a) £ M* x R 4 (where R* = (0 , oo)), and th e covariant derivativ e which corresponds t o the parameter s (A , a) i s im((x — ^dx 1 ) (1.16) V = d + with curvatur e (1.17) i? v = A5 (A2 + \x - a| 2 )' im(dx A dx l ) (A2 + | z - a | 2 ) 2 ' g) Covarian t derivative s an d connection s Before ending this first lecture, I would like to explain the relationshi p between covarian t derivative s an d connections . T o begin, conside r tha t V come s equipped wit h it s projectio n 7 r : V — X. Th e differentia l o f n induce s a sequenc e (1.18) Ver t - U TV - ^ TX where Ver t is , b y definition , th e kerne l o f dir. O f course , thi s kerne l i s generated by translations up and down the fiber of V, which means tha t (1.19) Ver t w TT* F Here, I have introduce d th e pull-bac k bundl e n*V. (I n general , i f Y i s a smoot h manifol d wit h a smoot h ma p / : Y — • X : the n / pulls-bac k V t o a vector bundl e (1.20) f*V = {(y,v) €YxV: f(y) = 7r(t ) } over Y. Tha t is , the fiber o f f*V a t a poin t y i s simply th e fiber o f V at /(i/). ) The sequenc e (1.21) ir*V - U T V ^UTX

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