From: "R. Bryant" <b1r2y3a4n5t@math.duke.edu>
Subject: Re: straight lines as geodesics
Date: 6 Sep 2000 08:30:03 -0500
Newsgroups: sci.math.research
Summary: Beltrami's theorem: if geodesics are straight lines, curvature essentially constant

Allan Adler wrote:

> I seem to remember reading somewhere that someone had classified (or at
> least tried to do so) all metrics on Euclidean space for which the geodesics
> are just straight lines. Does anyone know what paper that might be and
> whether similar work has been done for other natural families of curves?

    This is the 3-dimensional version of Beltrami's theorem, which is valid
in all dimensions and asserts that any metric on an open set in R^n whose
geodesics are the straight lines is projectively equivalent to a constant
multiple
of one of the three standard examples of constant curvature.  I don't know
the original reference, but, for example, see Spivak's 'A Comprehensive
Introduction to Differential Geometry", Vol 4, pg.23 and what follows.
   Spivak doesn't actually state the full theorem, but you can get it from
that.

    As for similar work, that depends on what you mean.  There is a test
for a 2-parameter family of curves in the plane to determine whether or
not they are the geodesics of a metric, and, similarly, there are tests in
higher dimensions (which, paradoxically, turn out to be much simpler
than the 2-dimensional case).  In general, pairs of metrics with the
same geodesics were classified by Darboux (see t. III of his "Theorie
Generale des Surfaces).  There is a treatment in English by Eisenhart,
but I don't know it well.   The upshot is that, for 'most' metrics on the
plane, the only metrics with the same geodesics are the constant
multiples of the given one.  Thus, the flat metric is very exceptional
in this regard.

    Yours,

    Robert Bryant