Editing Geodesic (section)

===Affine and projective geodesics===
Equation ({{EquationNote|1}}) is invariant under affine reparameterizations; that is, parameterizations of the form
:<math>t\mapsto at+b</math>
where ''a'' and ''b'' are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of ({{EquationNote|1}}) are called geodesics with '''affine parameter'''.

An affine connection is ''determined by'' its family of affinely parameterized geodesics, up to [[torsion tensor|torsion]] {{harv|Spivak|1999|loc=Chapter 6, Addendum I}}.  The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection.  More precisely, if <math>\nabla, \bar{\nabla}</math> are two connections such that the difference tensor
:<math>D(X,Y) = \nabla_XY-\bar{\nabla}_XY</math>
is [[skew-symmetric matrix|skew-symmetric]], then <math>\nabla</math> and <math>\bar{\nabla}</math> have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as  <math>\nabla</math>, but with vanishing torsion.

Geodesics without a particular parameterization are described by a [[projective connection]].