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  3. 4. let m" and n be smooth manifolds, tm and tn be

4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles....

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4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles. (a) Define what is a smooth vector field on M. (b) Let $2n-1 denote the (2n 1)-dimensional unit sphere in R2n for n > 2. Show that its tangent bundle TS2n-1 admits at least one smooth nowhere vanishing vector field X. (You may wish to try n = 2 first.) (C) Suppose TM and TN are both trivial. Show that T(MXN) is trivial. (Recall an l-dimensional smooth manifold P' has trivial tangent bundle TP if P admits / linearly independent smooth vector fields X1, ... , X1. Note that an example is M = N = St so that M N = T2, the torus.) 4. Let M" and N be smooth manifolds, TM and TN be their tangent bundles. (a) Define what is a smooth vector field on M. (b) Let $2n-1 denote the (2n 1)-dimensional unit sphere in R2n for n > 2. Show that its tangent bundle TS2n-1 admits at least one smooth nowhere vanishing vector field X. (You may wish to try n = 2 first.) (C) Suppose TM and TN are both trivial. Show that T(MXN) is trivial. (Recall an l-dimensional smooth manifold P' has trivial tangent bundle TP if P admits / linearly independent smooth vector fields X1, ... , X1. Note that an example is M = N = St so that M N = T2, the torus.)

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