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We give a classification for connected complete locally irreducibleRiemannian manifolds with nonpositive curvature operator, which admit a nonzeroclosed or co-closed conformal Killing $L^{2}-$form. Moreover, we provevanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms onsome complete Riemannian manifolds.

Conformal Killing $L^{2}-$forms on complete Riemannian manifolds with nonpositive curvature operator