TITLE: Locally constant sheaf but not constant
QUESTION [14 upvotes]: I am searching for a locally constant sheaf which is  not constant. Locally constant means that there exists an open covering of the total space such that the sheaf restricted to each open set in the cover is isomorphic to a constant sheaf. But the sheaf should not be constant throughout X. Does there exist such an example. Please help.

REPLY [2 votes]: Here's a concrete example from differential geometry: pick a flat Riemannian manifold $M$ whose tangent bundle is not isomorphic to a trivial bundle. For example, take $M$ to be the Mobius strip. Then the kernel of the Levi-Civita connection $d^{\nabla}: \mathcal{T}_M \to \Omega_M^{1} \otimes \mathcal{T}_M$ is the sheaf of parallel vector fields, which is a locally constant sheaf. It cannot be a constant sheaf, since the tangent bundle does not admit a global frame.