Curvature and Lagrangian submanifolds of the homogeneous nearly Kähler $\mathbb{C}P^3$

A tractable definition of the homogeneous nearly Kähler structure on $\mathbb{C}P^3$ is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on $\mathbb{C}P^3$, providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly Kähler $\mathbb{C}P^3$, and the nonexistence of Lagrangians with constant sectional curvature is established.