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SOLVED: 10-10. Suppose M is a compact smooth manifold and E → M is a smooth vector bundle of rank k. Use transversality to prove that E admits a smooth section σ with the following property: if k>dim M, then σ is nowhere vanishing; while if k ≤dim M, then the set of points where σ vanishes is a smooth compact codimension- k submanifold of M. Use this to show that M admits a smooth vector field with only finitely many singular points.
SOLVED: 10-10. Suppose M is a compact smooth manifold and E → M is a smooth vector bundle of rank k. Use transversality to prove that E admits a smooth section σ with the following property: if k>dim M, then σ is nowhere vanishing; while if k ≤dim M, then the set of points where σ vanishes is a smooth compact codimension- k submanifold of M. Use this to show that M admits a smooth vector field with only finitely many singular points.
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