Spin Real Projective Space Bundle

• Main Authors: Lin, Kuen-Huei, 林坤輝
• Other Authors: Cherng-Yih Yu
• Format: Others
• Language: zh-TW
• Published: 1998
• Online Access: http://ndltd.ncl.edu.tw/handle/83866743246338587974

碩士 === 淡江大學 === 數學學系 === 86 === In relation to the works of Kazdan and Warner, it is important todescribe the condition for a given manifold to admit a metric of positivescalar curvature. However, the spin structure plays an important role inthe question of the existence of metrics of positive scalar curvature. Inthe paper of Tetsuro Miyazaki, we see that he uses the complex projectivespace bundle and quaternion projective space bundle to represent a spinmanifold of positive scalar curvature. It is the reason that we discussthe spin real projective space bundle. In this thesis we want to find the rules which are convenient for us todetermine the real projective space bundle to be spin. In the first threechapters, we will discuss the smooth structure on manifolds, variousconstructions involving vector bundles and the Stiefel-Whitney classes ofa vector bundle. In the last chapter, we will introduce the spin structureon a vector bundle and find the necessary conditions for real projectivespace bundle to be spin. We have that the necessary condition for real projective space bundle tobe spin is the fiber''s dimension to be congruent to 3 mod 4 provided thereal projective space bundle whose dimension of fiber is greater than one. Moreover, we consider the following two cases to look at. nCase 1 : RP(α)=RP(mH⊕sε(RP )) Case 2 : RP(β)=RP(ap*H⊕b(L ⊙p*H)⊕cL⊕dε(RP(α)))Result : (1) If the fiber''s dimension of RP(α) is greater than one, then RP(α) is spin if and only if (m,s;n) is equal to (0'',0'',3''), (2'',2'',1''), (2,0,3''), (1,1,2'') or (0,2,3''). (2) The spin real projective space bundle RP(β) is constructible if and only if (a+b+c+d-1,m+s-1,n) is equal to (3'',3'',all), (3'',1'',all), (1,3'',all), (1,2'',1''), (1,2'',3''), (1,1,1''), (1,1,2'') or (1,1,3'') except for (3,3'',0'') and (3,3'',2''). (3) If RP(α) is spin and a+b+c+d-1 either congruent to 3 mod 4 or equal to one then RP(β) is spin.Note : ⊙ denotes tensor product and i'' means congruent to i mod 4 for i=1,2,3..