Hodge decomposition along the leaves of Riemannian foliation (Q1178634)

Let \(\mathcal F\) be a smooth foliation on a closed manifold \(M\). Suppose that \(M\) is equipped with a Riemannian metric, which decomposes \(TM\) as an orthogonal direct sum \(T{\mathcal F}\oplus Q\) of the tangent bundle \(T{\mathcal F}\) for the foliation and its normal bundle \(Q=TM/T{\mathcal F}\). This decomposition gives rise to a bigrading of the algebra of smooth differential forms \(\Omega\) on \(M\) and one can define a leafwise Laplacian, canonically associated to the exterior derivative along the forms on \(Q\). In the main theorem, a Hodge decomposition theorem is proved for this leafwise Laplacian. The theorem implies the usual Hodge decomposition on forms, when an ordinary closed Riemannian manifold is viewed as a single leaf foliation with zero normal bundle.