arXiv subjects
Publications and source records attributed to Jesus A. Alvarez Lopez.
For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.
For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of the corresponding eigenforms also leads to a Hodge theoretic description of this spectral sequence. This is an extension of results of Mazzeo-Melrose and R. Forman.
Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural Lie group $G$. Its transverse Lie structure can be considered as a transverse action $Φ$ of $G$ on $(M,\mathcal F)$; i.e., an ``action'' which is defined up to leafwise homotopies. This $Φ$ induces an action $Φ^*$ of $G$ on the reduced leafwise cohomology $\bar H(\mathcal F)$. By using leafwise Hodge theory, the supertrace of $Φ^*$ can be defined as a distribution $L_{dis}(\mathcal F)$ on $G$ called the Lefschetz distribution of $\mathcal F$. A distributional version of the Gauss-Bonett theorem is proved, which describes $L_{dis}(\mathcal F)$ around the identity element. On any small enough open subset of $G$, $L_{dis}(\mathcal F)$ is described by a distributional version of the Lefschetz trace formula.