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.
The easiest counterexample is the Riemannian product $$M=F\times N$$ for $F$ a manifold of constant non-zero sectional curvature and $N$ an arbitrary Riemannian manifold of dimension $n-k$. ($M$ is foliated by the copies of $F$.)
Any plane that is the product of a line in $F$ with a line in $N$ will be locally isometric to ${\mathbb R}^2$ and thus have sectional curvature equal to zero.
So the sectional curvature is not constant, even if the sectional curvature of $N$ were the same as that of $F$. (Which of course needn‘t be the case anyway.)
The Hopf fibration on $S^3$ with the standard round metric gives a counterexample (one dimensional hence flat leaves).
In dimension 2 (where sectional=Gauss), take for example any non flat metric on the torus foliated by circles.
One more counterexample: the hyperbolic $n$-space is foliated by horospheres with common center. Horospheres have zero curvature, the hyperbolic space curvature $-1$.
