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Rigidity results for complete manifolds with nonnegative scalar curvature

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\times \mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and further prove a rigidity result for the equality case.