  We study the classification of closed, smooth, spin, $1$-connected
$7$-manifolds whose integral cohomology ring is isomorphic to
$H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology
ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to
$H^*(\mathbb{C}P^2\times S^3)$ or $H^*(S^2\times S^5)$, this $7$-manifold
admits a Riemannian metric with positive Ricci curvature.
