I was assigned the task of calculating the rank of the complex projective space $\mathbb C P^n=SU(n)/S(U(1)\times U(n-1))$ and am not sure how best to approach that task. (looking in the classification of symmetric spaces does not count as calculating)
I considered using the fact that the sectional curvature is $$ sec(z,w)=1+3|Im \langle w,z\rangle|>0 $$ For any $z,w\in T_p(\mathbb C P^n)$ as proven in "Riemannian Geometry" by Peter Petersen (pp.222-224, Chapter 8.2.3) and then concluding that this implies that each totally geodesic submanifold has nonzero curvature and is therefore either one-dimensional or not flat.
I am not sure if that conclusion is correct and would like hints as to where I can find more information. If it is indeed incorrect, why is it? And what is a better way to prove it?