Spectral sequences and adiabatic limits (Q1842614)

Let \((M,g)\) be a compact Riemannian manifold. Suppose there is given an orthogonal splitting of the tangent bundle, \(\text{TM} = A \oplus B\). Write \(g = g_A \oplus g_B\). This yields a one-parameter family of metrics \(g_\delta = g_A \oplus \delta^{-2} g_B\) for \(0 < \delta \leq 1\). The limit of \((M,g_\delta)\) as \(\delta \to 0\) is known as ``adiabatic limit''. In the present paper a spectral sequence associated with \(A\) and \(B\) for the cohomology of \(M\) is studied. It is related to the asymptotic behaviour as \(\delta \to 0\) of the eigenvalues of the Laplace operator induced by \(g_\delta\) acting on \(p\)-forms. If \(A\) is integrable, then the spectral sequence is isomorphic to the standard Leray spectral sequence associated to the foliation \(A\).