Totally geodesic hypersurface on high dimensional sphere near round metric

Question

Let $g$ be a metric $C^2$-close enough to the round metric $g_0$. Can we still guarantee the existence of totally geodesic hypersurface for the metric $g$. Maybe some embedded $S^{n-1}$... I slightly know the existence of closed geodesic using min-max and seem like this cannot be extended to the higher dimensional sphere.