# Integration By Parts on Non-compact Manifolds

This is undoubtedly a very easy question, but perhaps there are some subtleties. Under what circumstances can we integrate by parts over a non-compact Riemannian manifold? I am aware that having bounded curvature is sufficient (is there a reference for this?), but can this be weakened?

• Integrate "what" by parts? Stokes theorem will hold provided whatever you are integrating converges "at infinity" in a suitable sense (see Deane's answer below), and does not require a Riemannian structure. So you can have a suitable equality of the form $\int \alpha \wedge \mathrm{d}\beta = \epsilon \int \mathrm{d}\alpha\wedge\beta$. For functions on a manifold you don't necessarily have natural partial derivatives. If you are thinking about directional derivatives relative to a vector field then you need the vector field to be parallel, other wise you get an error term. – Willie Wong Feb 29 '12 at 13:49
• The question could be rephrased "under what curcumstances do we have the formula [ \int_{\mathcal{M}} f \Delta g \, dV = \int_{\mathcal{M}} g \Delta f \, dV ]" – Michael Coffey Feb 29 '12 at 19:31