METRICS, CONNECTION S AN D GLUIN G THEOREM S 3 Here, p : ®2T*X A2T*X i s th e anti-symmetrizatio n homorphism . (In th e firs t equatio n above , g shoul d b e considere d t o b e a sectio n of ® 2 T*X.) I n genera l a connectio n ca n satisf y eithe r o r non e o f th e conditions i n (1.1) . The covariant derivativ e which satisfies (1.1 ) is called the Levi-Civit a covariant derivative . A covarian t derivativ e V o n a vecto r bundl e V naturall y define s a section of End(V) ® A2T*X, it s curvature. Thi s section, i?V , i s define d by the formula : (1.2) # v - S = p(V(V S )), where th e secon d V , a covarian t derivativ e o n V ® T*X, i s define d using th e origina l V wit h an y torsio n fre e connectio n o n T*X. I n th e case wher e V i s th e Levi-Civit a covarian t derivative , th e metri c ca n be use d t o defin e th e curvatur e R v a s a sectio n o f A 2 T*X ® A 2 T*X. Alternately, fo r th e Levi-Civit a covarian t derivative s on e ca n thin k o f Rv a s a bilinear form on A2TX\ i n this guise, R v i s always a symmetri c bilinear form . So ends the introductio n t o Riemannia n geometry . Th e res t o f thes e lectures concern s the specializations whic h occu r whe n X i s a 4-dimen - sional, oriented manifold . b) A brie f revie w o f anti-sel f dualit y Anti-self dualit y owe s it s ver y existenc e t o th e basi c fac t tha t th e Lie algebr a so(4 ) o f th e specia l orthogona l grou p SO(4) i s no t simple , rather thi s Li e algebr a decompose s a s th e direc t su m o f tw o copie s o f the Li e algebra so(3) o f the grou p 50(3) : (1.3 ) so(4 ) ^so(3)0so(3) . (On th e grou p level , thi s decompositio n i s a manifestatio n o f th e fac t that th e universa l coverin g group of SO (4) is the product o f two copie s of 5/7(2), the group of 2 x 2, unitary matrices over the complex number s with determinan t equa l t o 1. ) The decomposition of so (4) has the following geometric consequences: Let X denot e a smooth , oriented , 4-dimensiona l manifold . A choic e of metric, g, o n TX define s a direc t su m decompositio n (i.4) A 2 n ^ A + e A - as a sum of two oriented 3-plan e bundles. Th e bundles A ± ar e describe d as follows: Le t {e 1 , e2, e3, e4} be an oriented orthonormal fram e fo r T*X
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