  We study, on a weighted Riemannian manifold of Ric$_{N} \geq K > 0$ for $N <
-1$, when equality holds in the isoperimetric inequality. Our main theorem
asserts that such a manifold is necessarily isometric to the warped product
$\mathbb{R} \times_{\cosh(\sqrt{K/(1-N)}t)} \Sigma^{n-1}$ of hyperbolic nature,
where $\Sigma^{n-1}$ is an $(n-1)$-dimensional manifold with lower weighted
Ricci curvature bound and $\mathbb{R}$ is equipped with a hyperbolic cosine
measure. This is a similar phenomenon to the equality condition of Poincar\'e
inequality. Moreover, every isoperimetric minimizer set is isometric to a
half-space in an appropriate sense.
