**Fact** : Consider the inclusion $V^{n-1} \rightarrow M^n$ where $M$ is a closed orientable simply
connected positively curved manifold.

Then connectivity lemma implies that the inclusion is $(n-1)$-connected so that $M$ is homeomorphic to a sphere.

**Situation** : As you know ther exists a $S^3$-action on $M={\bf CP}^2$ which is cohomogeneity one, i.e., $M/S^2=[0,1]$.

Hence $M$ is the union of two disk bundles over two singular orbits $S^3\cdot x_1$, $S^3\cdot x_2$ : $S^3\cdot x_1$ is diffeomorphic to $S^2$ and $ S^3\cdot x_2 = \{ x_2\}$.

Here think about two conditions :

(1) A geodesic sphere of suitable radius around $x_2$ is totally geodesic.

(2) $M$ is positively curved.

By the above fact (1) and (2) cannot be compatible.

Here I have a question : Is it true that cannonical $S^5(1)/S^1={\bf CP}^2$ does not have a codimensional 1 totally geodesic submanifold.

Thank you in advance.