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On the harmonic and Killing tensor field on a compact Riemannian manifold. (English) Zbl 1038.53037
The authors study the vector spaces \(H^q(M,R)\) and \(K^q(M,R)\) of harmonic \(q\)-forms and Killing exterior \(q\)-forms on a compact Riemannian manifold \(M\) without boundary. For any \(q\)-form \(w\) on \(M\), the authors introduce a quadratic form \(\widehat {F}_q(w,w)\), whose nullity is a global property when \(M\) is a compact Riemannian manifold of dimension \(n\geq 3\). Then the following results are proved: Assume that the quadratic form \(\widehat {F}_q(w,w)\) is semi-positive (resp. semi-negative) on \(M\) and the nullity of \(\widehat {F}_q(w,w)\) is equal to the number of linearly independent parallel exterior \(q\)-forms. Then we have \(b_q(M)=\text{nullity }(\widehat {F}_q(w,w))\) (resp. \(\dim K_q(M,R)=\text{nullity }(\widehat {F}_q(w,w)\)), where \(b_q(M)\) is the \(q\)th Betti number of \(M\).
MSC:
53C20 Global Riemannian geometry, including pinching
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