  The Hodge decomposition is well-known for compact manifolds. The result has
been extended by Kodaira to include non-compact manifolds and $L^2$ forms. We
further extend the Hodge decomposition to the Sobolev space $H^1$ for general
$k$-forms on non-compact manifolds of nonpositive constant sectional curvature.
As a result, we also obtain a decomposition on $\mathbb R^N$.
