What are the umbilic hypersurfaces in a sphere? It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersu...--prophetes.ai

What are the umbilic hypersurfaces in a sphere? It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersurfaces of a constant curvature sphere $S^n$? I assume that these are exactly the intersections of $S^n$ with hyperplanes of $\mathbb{R}^{n+1}$ (i.e. geodesic spheres around some point), but I wasn't able to find a proof. Can someone please give me a reference?

Spivak's comprehensive introduction to Differential Geometry, Vol 4 2n edition, page 112.