differential geometry w notes from teacher_Part_97

differential geometry w notes from teacher_Part_97 - 193...

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7.5. HARMONIC FORMS 193 The Hodge Laplacian on p -forms is the operator L : C ( Λ p ( M )) C ( Λ p ( M )) defined by L = d δ + δ d = ( d + δ ) 2 . The operators d and δ commute with the Hodge Laplacian, i.e. dL = Ld , δ L = L δ . Theorem 7.5.3 For any p there holds L = Δ + W , where W : Λ p Λ p is an endomorphism on the bundle of p-forms called the Weitzenb¨ock endomorphism . Proof : 1. ± Weitzenb¨ock endomorphism is a linear combination of Riemann curvature tensor, that is, when acting on p -forms W has the form W i 1 ... i p j 1 ... j p = F mni 1 ... i p kl j 1 ... j p R kl mn , where F mni 1 ... i p kl j 1 ... j p is constructed only from the Kronecker symbol δ i j and the metric g i j and g i j . Problem . Obtain the expression for the Weitzenb¨ock endomorphism for p -forms. Hint: replace partial derivatives by covariant derivatives and use the definition of the curvature . A p -form α is called harmonic if L α = 0 . di geom.tex; April 12, 2006; 17:59; p. 191
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194 CHAPTER 7. HOMOLOGY THEORY Theorem 7.5.4 Let M be a closed Riemannian manifold. Then a p- form α is harmonic if and only if it is closed and coclosed, that is,
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