4 C. T A U B E S over a n ope n se t U C X. The n (1.5) = Spa n (e1 A e2 ± e 3 A e4), (e2 A e3 ± e 1 A e4), (e3 A e1 ± e 2 A e4). (Here i s a secon d vie w o f A ± \ Th e choic e o f metri c g define s a re - duction o f the principal GL(4 E) bundl e of frames i n TX t o a principa l 50(4) bundle , P r X o f orthonormal , oriente d frames . Th e grou p SO(4) ha s two distinct representation s i n BO (3), these correspond to th e two possible projections o f 50(4)'s adjoin t representatio n ont o the tw o factors o f so (3) i n (1.3) . The n A* 1 ca n b e viewed a s vector bundle s wit h fibers s o (3) whic h ar e associate d t o th e 50(4 ) principa l bundl e P r b y the th e aforementione d tw o representations. ) The decompositio n i n (1.4 ) i s covariantl y constan t i n th e sens e tha t the g' s Levi-Civit a covarian t derivativ e o n A 2 T*X map s section s o f A ± into sections of ®T*X for example, there is no mixed term mappin g A+ int o A ~ ® T*X. Remember fro m th e previou s sectio n ho w th e curvatur e o f the Levi - Civita connectio n define s a sectio n o f A 2 T*X ® A2T*X. Well , writ e A2T*X a s i n (1.4 ) an d ther e i s a resultin g decompositio n o f A 2 T*X g A2T*X int o fou r 3 x 3 blocks . Th e curvatur e R v o f th e Levi-Civit a connection split s a s follows wit h respec t t o thi s decomposition : W+ - s / B* \ B W--S-I))' Here, s i s 1/1 2 o f the scala r curvature , B i s isomorphic t o th e traceles s Ricci tensor, an d ar e the self dual an d anti-sel f dua l Weyl curvatur e tensors. A rathe r canonica l differentia l geometr y proble m i s t o find metric s on a give n manifol d wit h som e piec e o f (1.6 ) prescribed . Fo r example , the prescriptio n tha t s = constan t i s know n a s th e Yamab e problem , and was solved b y the work of Aubin [8 ] and Schoe n [42] . The prescrip - tion tha t B vanis h i s known a s Einstein's equation s (wit h cosmologica l constant). Th e prescriptio n tha t W + an d W_ bot h vanis h assert s tha t the metri c i n questio n i s conformall y flat. Tha t is , ever y poin t ha s a neighborhood wit h coordinates tp : R4 X whic h pull back the metri c as f gEi where / i s a positiv e functio n an d wher e g E i s the Euclidea n metric o n E 4 . c) Th e questio n o f W + = 0 A novelty of 4 dimensions is that on e can as k for metric s on X whic h have W+ = 0 . I n general, one knows next to nothing about this question. In fact , ther e i s a rather shor t lis t o f manifolds wit h suc h metrics . Th e (1.6) Rv•v=_
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