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differential geometry w notes from teacher_Part_96

differential geometry w notes from teacher_Part_96 - 191...

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7.5. HARMONIC FORMS 191 The coderivative is a linear map δ : Λ p Λ p - 1 defined by δ = * - 1 d * = ( - 1) ( n - p + 1)( p - 1) * d * . The exterior derivative and the coderivative satisfy the important conditions d 2 = δ 2 = 0 . Problem . Show that in local coordinates the coderivative of a p -form α is the ( p - 1)-form δα with components ( δα ) i 1 ... i p - 1 = g i 1 j 2 · · · g i p - 1 j p 1 | g | j | g | g jk 1 g j 2 k 2 · · · g j p k p α k 1 k 2 ... k p Now we define the L 2 -inner product of p -forms by ( α, β ) L 2 = M α ∧ * β = M α, β vol , and the L 2 -norm || α || L 2 = ( α, α ) L 2 . This makes the space C ( Λ p ( M )) of smooth p -forms an inner-product vec- tor space. The completion of C ( Λ p ( M )) in the L 2 -norm gives the Hilbert space L 2 ( Λ p ( M )) of square-integrable p -forms. Let A : H H be an operator on a Hilbert space H . The adjoint of the operator A with respect to the inner product of the space H is the operator A * : H H defined by, for any ϕ, ψ H , ( A ϕ, ψ ) = ( ϕ, A * ψ ) . di ff geom.tex; April 12, 2006; 17:59; p. 189

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192 CHAPTER 7. HOMOLOGY THEORY Theorem 7.5.1 Let M be a closed orientable Riemannian manifold. Then the adjoint of the exterior derivative is the negative coderivative
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