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Joachim Lohkamp has shown (Annals of Mathematics, 1994Annals of Mathematics, 1994) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.
Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.
Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.
Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes.
