The graph of a totally geodesic foliation

Abstract

We study the properties of the graph of a totally geodesic foliation. We limit our considerations to basic properties of the graph, and from them we derive several interesting corollaries on the structure of leaves. In this short note we study the properties of the graph of a totally geodesic foliation. Its importance comes from the fact that the graph of a foliation is the starting point of the construction of the C∗-algebra associated with this foliation. The value of C∗-algebras in the study of foliations cannot be overestimated, but this goes beyond the scope of this note. We limit our considerations to basic properties of the graph, and from them we derive several interesting corollaries on the structure of leaves. The definition and the basic properties of the graph of a foliation can be found in [8]. Let us recall the definition. The graph GR(F) of the foliation F is the space of equivalence classes of triples (y, α, x) where x and y are points of the same leaf L of F and α is a path in L linking x to y. Two triples (y, α, x) and (y′, α′, x′) are equivalent iff x = x′, y = y′ and the holonomy of the curve α−1 ∗ α′ is trivial. A neighbourhood of 〈y, α, x〉 consists of elements represented by the triples of the form (y′, α′, x′) where x′ belongs to some neighbourhood of x in a transverse manifold passing through x, y′ belongs to some neighbourhood of y in a transverse manifold passing through y, and α′ is the holonomy lift of α to x′ (see [8]). In the same paper it is proved that the graph of a foliation is a manifold of dimension n + p but in general non-Hausdorff. Moreover, if the elements of the holonomy pseudogroup are determined by their jets then the graph is a Hausdorff topological space. The mappings p1 : 〈y, α, x〉 7→ x, 1991 Mathematics Subject Classification: Primary 53C12, 57R30.