I was reading about totally geodesic hypersurfaces when I found the next proposition:
Proposition: The sectional curvature $K$ of $M$ is constant at $p$ if and only if every unit vector in $T_{p}M$ is normal to a totally geodesic hypersurface at $p$.
The proof is following by Codazzi equation.
I'm stuck proving this.
If $K$ is constant, we get from Codazzi equation that $R_{xy}x=K(\langle x,x\rangle y-\langle x,y\rangle x).$ Then,for nonnull $x\perp y$ such equation becomes $R_{xy}x=\langle x,x\rangle K(x,y) y.$ But I don't get how this works to prove that each unit vector on $T_{p}M$ is normal to a totally geodesic hypersurface. I know that a semi-Riemann submanifold is totally geodesic if the shape tensor vanishes: $\mathrm{II}=0$ but I can't see how this works with the above to get the desire result.
For the other direction I'm not sure how to proceed to get that $K$ is constant.
Any kind of help is thanked in advanced.