Tangent and Cotangent Bundles
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چکیده
of subsets of TM: Note that i) 8 (p;Xp) 2 TM , as p 2M ) there exists (U ; ) 2 S such that p 2 U ; i.e. (p;Xp) 2 TU , and we have TU = 1 (R) 2 : ii) If we de ne F : TpM ! R by F (Xp) = (Xp(x); Xp(x); :::::; Xp(x)) where x; x; ::::; x are local coordinates on (U ; ), then clearly F is an isomorphism, so (p; Xp) = ( (p); F ( Xp)); and 1 = ( 1 ; F 1 ): Now take 1 (U); 1 (V ) 2 and suppose (p; Xp) 2 1 (U)\ 1 (V ) for some U; V open in R: ) (p; Xp) 2 U and (p; Xp) 2 V Take U = U U and V = V V where U ; U ; V ; V are open sets in R:Clearly (p) 2 U ; F ( Xp) 2 U ; (p) 2 V , and F (Xp) 2 V : From the de nition of open set there exist neighborhoodsW 1 ;W 2 ;W 1 ;W 2 of (p); (p); F ( Xp) and F (Xp) respectively such that
منابع مشابه
Natural tensor fields of type (0, 2) on the tangent and cotangent bundles of a Fedosov manifold
To any (0,2)-tensor field on the tangent and cotangent bundles of a Fedosov manifold, we associate a global matrix function ‘mutatis mutandis’ as in the semi-Riemannian case. Based on this fact, natural (0,2)-tensor fields on these bundles are defined and characterized. Mathematics Subject Classification: 53C99; 53C50.
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