So as the title says; Why the sphere $S^2$ can't be embedded as a totally real submanifod of $\mathbb CP^2$? I asked a similar question about totally real submanifolds. e.g see Totally real submanifold of 2 dimensional complex manifold
Number 3 of the answer mentions the proof of $S^2$ being Lagrangian. But if $S^2$ is Lagrangian in $\mathbb CP^2$ then it follows that $S^2$ is a totally real submanifold! Is it actually?