Rectification of circles, spheres, and classical geometries

grantor: University of TorontoThe goal of this thesis is to describe all local diffeomorphisms mapping a collection of circles (res. spheres) in a neighborhood into straight lines (res. hyperspaces). This description contains two main results. The first is a complete description of the general rectifiable collection of circles in 'R'3 (or spheres of codimension 1 or 2 in ' Rn' passing through one point. It turns out that to be rectifiable all circles (or all spheres) need to pass through some other common point. The second main result is a complete description of geometries in which all the geodesics are circles. This is a consequence of an extension of the Beltrami theorem by replacing straight lines with circles.Ph.D