Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections.
Given a de Rham cohomology class $[\omega] \in H^k(M; \mathbb{R})$ and a parallel global section $\sigma \in H^0(M; \mathcal{L})$, what methods can I use to check if the cohomology class (with local coefficients) $[\omega \otimes \sigma] \in H^k(M; \mathcal{L})$ is non-zero?
If that is too broad a question, let's say that I start with a parallel global section $\sigma$ of my vector bundle and I wish to find a de Rham cocycle $\omega$ that will give me a non-trivial class $[\omega \otimes \sigma]$ in cohomology with local coefficients. I know in general this will not necessarily be possible to do, since the higher cohomology groups with local cofficients might vanish. But are there some heuristics I could use help me? For example, taking $\omega$ to be some non-vanishing Stiefel-Whitney or Pontryagin class, etc.
For example, if I understand correctly, I believe that in hyperbolic geometry we can use the Poincaré dual of a totally geodesic hypersurface to build degree 1 cohomology classes with local coefficients, at least for certain representations. I don't know if this works with totally geodesic submanifolds of other codimensions, but that would be nice if it did!
Apologies if this is a bit elementary; I'm not very experience with algebraic topology (though I'm reasonably familiar with category theory and sheaf theory).