Qualifying Examination
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چکیده
Solution: Since V is locally trivializable and M is compact, one can find a finite open cover Ui, i = 1, . . . , n, of M and trivializations Ti : V |Ui → Rk. Thus, each Ti is a smooth map which restricts to a linear isomorphism on each fiber of V |Ui . Next, choose a smooth partition of unity {fi}i=1,...,n subordinate to the cover {Ui}i=1,...,n. If p : V → M is the projection to the base, then there are maps V |Ui → R, v 7→ fi(p(v))Ti(v) which extend (by zero) to all of V and which we denote by fiTi. Together, the fiTi give a map T : V → Rnk which has maximal rank k everywhere, because at each point of X at least one of the fi is non-zero. Thus V is isomorphic to a subbundle, T (V ), of the trivial bundle, Rnk. Using the standard inner product on Rnk we get an orthogonal bundle W = T (V )⊥ which has the desired property. For the second part, embed S2 into R3 in the usual way, then
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تاریخ انتشار 2016