The Fermi Flow and its Application to Geometry

We introduce the notion of Fermi flow for hypersurfaces in Riemannian manifolds. It turns out that this is a powerful tool to study the geometry of distance surfaces about a given initial hypersurface. Some of the results in this paper are known in one form or another, however the aim is to demonstrate how they can all be derived by the same method and proved in a very simple manner. In addition we obtain some new results and results that are stronger than those stated in the literature.