Hi all! I have a possibly trivial (possibly non-trivial? :rofl:) question. Here it is: Assumption-Assume I have a closed p-form, whose integral over any p-cycle is always zero. Statement-The closed p-form is also exact, by what is sometimes called de Rham's first theorem My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??) Further question: de Rham's theorem is often proved for compact manifolds. Is my statement true even for manifolds which are not compact? (assume my manifold is paracompact, but not compact). Thanks a lot!