Elmer Rees

The existence of bitangent spheres

Given any point x on a smooth, closed submanifold M of Euclidean space, there is a bitangent hyperplane or sphere that is tangent to M at x .

Fibre singularities of functions on principal S1-bundles

Abstract Let π : E → B denote a smooth principal S 1 -bundle with total space E and base space B of dimension m . In C.Z.W. Hassel Sweatman, 1993, it was noted that, as a consequence of the general results of K. Igusa, 1990, the set of smooth real-valued functions on E whose singularities on fibres are only of types A 1 ,…, A m + 1 is open and dense in the Whitney C ∞ topology and further results on the existence of good functions were proved there. In this article, we prove the existence of a smooth real-valued function on E with no fibre singularities of type A k for k > [ m /2] + 1.