Classification for smooth manifolds looking like CP³× S⁷

In this paper, we classify simply connected closed smooth $13$-dimensional manifolds whose cohomology ring is isomorphic to that of CP³× S⁷, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a manifold satisfies certain conditions, either itself or its connected sum with an exotic $13$-sphere Σ¹³ admits a Riemannian metric of non-negative sectional curvature. As an additional application of our classification, we classify the diagonal S¹-actions on S⁷× S⁷.