MANIFOLDS WITH TRANSVERSE FIELDS IN EUCLIDEAN SPACE

Abstract

This chapter presents the connection between differentiate structures on M, where M is a topological n-manifold imbedded in Euclidean (n + k)-space Rn+k, and transverse fields on M. A manifold with a differentiate structure may be differentiably imbedded in Euclidean space; therefore, the image manifold automatically possesses a transverse field. When the converse of this is seek, the main result states that if M is a combinatorially-triangulated manifold rectilinearly imbedded in Euclidean space, and M possesses a transverse field φ, then M possesses a differentiate structure M(φ) compatible with the triangulation. Two transverse fields that are homotopic induce differentiate structures that are diffeomorphic. In addition, the chapter presents the problem of constructing a transverse field on a combinatorial n-manifold rectilinearly imbedded in a Euclidean space.