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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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