Given a $k$- dimensional foliation $F$ of a riemannian $n$-manifold $M$, with the property that the leaves of the foliation have constant sectional curvature $s$, for some $s$, is it true that $M$ will also have the same constant sectional curvature?

Is the same true if sectional curvature is replaced by the Gaussian curvature?

If it's a well known result, any hint at proving this or a possible reference or a counter example otherwise, will be most welcome.

Thanks.