This is a quick question, I'm not asking for a proof (unless there's a simple argument behind it), although maybe a reference would be nice to complement the answer.
Consider a riemannian manifold with dimension greater than $2$. I believe I remember that the weaker statement of sectional curvature being constant in each point implies the stronger one that the manifold has constant curvature.
Is this actually true or am I misremembering? Of course this is clearly not true for $n=2$ as in this case the sectional curvature is just a number in each point and does not imply the manifold has constant curvature.