This question is inspired by question in reference.

**Question :** If $M$ is a simply connected closed Riemannian manifold of nonnegative sectional curvature, then there is a totally geodesic submanifold $S$ of codimension 1

**Def :** ${\rm conv}\ X$ is a smallest closed convex set containing $X$.
And a subset $X$ is **convex** if for $x,\ y\in X$, any shortest geodesic between them is in $X$.

**Construction of $S$ :** Let $X_1=B(p,\delta)$ to be a closed ball, which is convex. If $p_1$ is not in $X_1$, then let $X_2={\rm conv}\ X_1\bigcup\{p_1\}$ s.t. $d_H(X_1,X_2) >0$ is small and ${\rm vol}\ X_2-{\rm vol}\ X_1 >0$ is small. Hence if we repeat this process, then $X_n$ goes to $ X_\infty$.

If $p_\infty$ is not in $X_\infty$, and $d_H(X_\infty,{\rm conv}\ X_\infty\bigcup\{p_\infty\})$ is large, then $\partial X_\infty$ is a desired one.

If not, we let $X_1=X_\infty$ and repeat these process. That is, ${\rm vol}\ X_\infty$ is *supremum of volumes of convex sets containing* $p$.

Example : If we starts this process at a point of negative sectional curvature in torus of revolution, then $X_\infty$ is closure of set of all points of negative sectional curvature.

Additional question :We have alvarezpaiva's answer. But I want to know in dimension $3$. Or I want to know why my argument fails in $\mathbb{C}P^2$.

**Reference :** Convex sets in Alexandrov spaces