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4.1. Let X be a smooth submanifold of R k and let ∆ = {(x, x) : x ∈ X} be the diagonal in X × X. Then the normal bundle to ∆ in X × X is defined by N(∆) = {(y, w) : y ∈ ∆, w ∈ Ty(X × X) and w · v = 0 for all v ∈ Ty(∆)} where · denotes the dot product in R k , and the projection π : N(∆) → ∆ is given by π(y, w) = y. Prove that N(∆) is equivalent to the tangent bundle T X. 8. Prove the following coordinate-free formula for exterior derivative on 1-forms: If ω is a differential 1-form on a smooth manifold M and X, Y are smooth vector fields on M, then dω(X, Y ) = Xω(Y ) − Y ω(X) − ω([X, Y ]). (Hint: Since both sides are linear in ω, it suffices to prove the result when ω = g df where f, g : M → R are smooth functions.) 11. (Cup products) Let α and β be closed forms on a smooth manifold M. Prove the following: (a) α ∧ β is closed. (b) If, in addition, α or β is exact then α ∧ β is exact. (c) Deduce that there is a well-defined product map (called the cup product) ∪ : H p (M) × H q (M) → H p+q (M) defined by [α]∪[β] = [α∧β] where α and β are closed forms whose cohomology classes are [α] and [β] respectively. (d) Show that if f : M → N is a smooth map then f ∗ (a ∪ b) = f ∗a ∪ f ∗ b for all a ∈ Hp (N), b ∈ Hq (N). 4. 2. Define S 2 to be the unit sphere in R 3 . For x ∈ S 2 and u, v ∈ TxS 2 prove that ω(u, v) := x · (u × v) is a closed differential 2-form on S 2 . Is it exact?

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