Department of Mathematics


Foliations on Manifolds

tom_maths

 

In collaboration with researchers in Switzerland and the United States of America, researchers at The University of Auckland are studying foliations of topological manifolds, especially manifolds which are so large that it is impossible to impose a metric on them.

A manifold is a topological space which, to a sufficiently short-sighted observer, looks like Euclidean space of some dimension: in dimension 1 it looks locally like a line, and examples include a line itself and a circle, in dimension 2 it looks locally like a very thin sheet of paper and examples include the surfaces of a sphere, a torus and the Klein bottle.

A foliation of a manifold is a subdivision of the manifold into what looks locally like parallel lines or surfaces and so on (called leaves), somewhat like the flow lines of a river or the pages of a paperback book which has been bent.

The illustration shows the first few steps in the construction of a rigid foliation of a disc, which is a foliation such that the only transformations of the disc which preserve the leaf structure map each leaf to itself.

Currently we are investigating foliations on non-metrisable manifolds. The simplest non-metrisable surface is the long plane which is like an ordinary plane except that it is so long in all directions that it cannot support a distance function (metric). To our amazement there are essentially only six different foliations of the long plane, in great contrast to an ordinary plane which supports infinitely many

Researchers at The University of Auckland