TITLE: Sectional curvature of leaves of foliation
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: One more counterexample: the hyperbolic $n$-space is foliated by horospheres with common center. Horospheres have zero curvature, the hyperbolic space curvature $-1$.