We study, on a weighted Riemannian manifold of $$\hbox {Ric}_{N} \ge K > 0$$ Ric N ≥ K > 0 for $$N < -1$$ N < - 1 , when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product $${\mathbb {R}} \times _{\cosh (\sqrt{K/(1-N)}t)} \Sigma ^{n-1}$$ R × cosh ( K / ( 1 - N ) t ) Σ n - 1 of hyperbolic nature, where $$\Sigma ^{n-1}$$ Σ n - 1 is an $$(n-1)$$ ( n - 1 ) -dimensional manifold with lower weighted Ricci curvature bound and $${\mathbb {R}}$$ R is equipped with a hyperbolic cosine measure. This is a similar phenomenon to the equality condition of Poincaré inequality. Moreover, every isoperimetric minimizer set is isometric to a half-space in an appropriate sense.