We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss–Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by K the nonzero constant Gauss–Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products S n − 1 a × S 1 1 − a 2 , a 2 = 1 / 1 + K 2 n − 2 or S n − 1 a × H 1 − 1 + a 2 , a 2 = 1 / K 2 n − 2 − 1 . $$ {S}^{n-1}(a)\times {S}^1\left(\sqrt{1-{a}^2}\right),\kern0.5em {a}^2=1/\left(1+{K}^{{\scriptscriptstyle \frac{2}{n-2}}}\right)\mathrm{or}\kern0.5em {S}^{n-1}(a)\times {H}^1\left(-\sqrt{1+{a}^2}\right),\kern0.5em {a}^2=1/\left({K}^{{\scriptscriptstyle \frac{2}{n-2}}}-1\right). $$