On Rectification of Circles and an Extension of Beltrami's Theorem

Details On Rectification of Circles and an Extension of Beltrami's Theorem On Rectification of Circles and an Extension of Beltrami's Theorem

2003-01-20

arxiv.org/abs/math/0301220v1

Mathematics

Differential Geometry

18 pages

Scientific paper

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be rectifiable all circles need to pass through some other common point. The second main result is a complete description of geometries in $\r^3$ in which all the geodesics are circles. This is a consequence of an extension of Beltrami's theorem by replacing straight lines with circles.}

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