In this paper, we show that
n-dimensional (
n ≥ 2) complete and non-compact smooth metric measure spaces with non-negative weighted Ricci curvature in which some Gagliardo-Nirenberg-type inequality holds are not far from the model metric measure
n-space (i.e., the Euclidean metric
n-space). Moreover, this fact, together with two generalized volume comparison theorems given in [P. Freitas
et al. Calc. Var. Partial Differential Equations
51 (2014), 701-724], surprisingly leads to an interesting rigidity theorem for the given metric measure spaces.
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