2 C. TAUBE S Riemannian metri c analog y o f th e anti-sel f dua l equations . Wit h thi s said, i t i s time t o start . a) A brie f revie w o f Riemannia n geometr y A manifol d X ha s a tangen t bundle , TX, an d a cotangen t bundl e T*X. Th e second exterior power of the cotangent bundle will be denoted by A2T*X. Th e third exterior power of T*X wil l be denoted by A 3 T*X, etc. Thes e exterio r power s of T*X ar e eac h oriente d naturall y whe n X is oriented . Now, le t g b e a Riemannia n metri c o n X. Thi s g is , pointwise , a Euclidean metri c o n TX whic h varie s smoothl y ove r X. Th e metri c g induce s a metri c o n T* X an d o n al l bundle s whic h ar e constructe d functorally fro m TX an d T*X (i.e. , th e exterio r an d tenso r power s o f T*X). Thi s happen s i n any dimension an d th e reade r i s referred t o an y standard tex t o n Riemannia n geometr y (i.e. , [32]) . Next, the reader shoul d recall the definition o f a covariant derivative : A covariant derivativ e o n a vector bundl e V X is , by definition, a n R-linear map , V, from section s of V t o sections of V®T*X whic h obey s V(/• s) = / Vs + s®df wheneve r / i s a function o n X an d s is a section of V. (Here , d is the usua l exterio r derivative. ) A covarian t derivativ e on V induce s a covarian t derivativ e (als o calle d V ) o n F' s dual , V* = Hom(V, R). Thi s covarian t derivativ e i s characterize d uniquel y b y th e following criteria : Le t t b e a sectio n o f V * an d le t s b e a sectio n o f V. The n d(t(s)) = (Vt)(s) + t(S7s). (Th e notatio n her e an d belo w is o n th e slopp y side , bu t I wil l continu e unappologeticall y an d refe r the confuse d reade r t o a standar d differentia l geometr y text. ) And , covariant derivative s o n a pai r o f bundle s V an d V induc e covarian t derivatives on their tenso r product V ® V an d direc t su m V © V i n th e most obviou s way . In particular , a covarian t derivativ e o n TX induce s one on T*X an d on al l o f th e tenso r bundle s ove r X\ thes e bein g direc t sum s o f tenso r products o f copie s o f TX an d T*X. Given a metri c g on th e tangen t bundle , ther e i s a uniqu e covarian t derivative o n thi s sam e bundl e whic h i s bot h metri c compatibl e an d torsion free . Tha t is , ther e i s a uniqu e covarian t derivative , V , whic h has th e followin g properties : 1) V # = 0 (Metri c compatibility ) 2) p(Vs) = ds (Torsio n free ) (i.i)
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