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We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\\geq 2$ has at least two ends and\n  \\[\n  \\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})\\geq 0,\n  \\]\n  for some $\\gamma<\\frac{4}{n-1}$, then $M$ splits isometrically as $\\mathbb R\\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. 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We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\\geq 2$ has at least two ends and\n  \\[\n  \\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})\\geq 0,\n  \\]\n  for some $\\gamma<\\frac{4}{n-1}$, then $M$ splits isometrically as $\\mathbb R\\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. 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