Foliations and Heat Diffusion

Abstract

This chapter is devoted to the theory of foliations. We briefly present the definition of a foliation and illustrate it by a number of examples (product foliation, a foliation given by a submersion, Reeb foliation of a solid torus, a linear foliation of a torus). We also recall the notion of the holonomy of a leaf. We present the foliated Laplace operator and foliated heat diffusion operators semigroup together with the definition of harmonic measures which plays essential role in the metric diffusion. We only demonstrate these facts which are necessary for further results. For the complete theory one can refer to Candel and Conlon (Foliations I & II. American Mathematical Society, Providence, 2001 & 2003), one of the best books about foliations. We also take advantage of some results from Moerdijk and Mrcun (Introduction to foliations and lie groupoids. Cambridge University Press, Cambridge, 2003), Tamura (Topology of foliations: an introduction. American Mathematical Society, Providence, 1992), and Walczak (Dynamics of foliations, groups and pseudogroups. Birkhäuser, Boston, 2004).