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Additional info for Diferentiable Manifolds - Section C Course 2003 (Lecture Notes)
Ip dxk ∧ dxj ∧ dxi1 ∧ dxi2 . . ∧ dxip . ip ∂xj ∂xk is symmetric in j, k but it multiplies dxk ∧dxj in the formula which is skew-symmetric in j and k, so the expression vanishes identically and d2 α = 0 as required. For the third part, we check on decomposable forms α = f dxi1 ∧ . . ∧ dxip = f dxI β = gdxj1 ∧ . . ∧ dxjq = gdxJ and extend by linearity. So d(α ∧ β) = = = = = d(f gdxI ∧ dxJ ) d(f g) ∧ dxI ∧ dxJ (f dg + gdf ) ∧ dxI ∧ dxJ (−1)p f dxI ∧ dg ∧ dxJ + df ∧ dxI ∧ gdxJ (−1)p α ∧ dβ + dα ∧ β So, using one coordinate system we have defined an operation d which satisfies the three conditions of the theorem.
Proof: Represent a ∈ H p (N ) by a closed p-form α and consider the pull-back form F ∗ α on M × [0, 1]. We can decompose this uniquely in the form F ∗ α = β + dt ∧ γ 50 (13) where β is a p-form on M (also depending on t) and γ is a (p−1)-form on M , depending on t. In a coordinate system it is clear how to do this, but more invariantly, the form β is just Ft∗ α. To get γ in an invariant manner, we can think of (x, s) → (x, s + t) as a local one-parameter group of diffeomorphisms of M × (a, b) which generates a vector field X = ∂/∂t.
Xn , y1 , . . , yn ) and on two overlapping coordinate charts we there had x1 , . . , x˜n , Φβ Φα−1 (x1 , . . , xn , y1 . . , yn ) = (˜ j 36 ∂ x˜i yi , . . , ∂x1 i ∂ x˜i yi ). ∂xn For the p-th exterior power we need to replace the Jacobian matrix J= ∂ x˜i ∂xj by its induced linear map ΛpJ : Λp Rn → Λp Rn . It’s a long and complicated expression if we write it down in a basis but it is invertible and each entry is a polynomial in C ∞ functions and hence gives a smooth map with smooth inverse.