  It is a standard result that the integral curves of an auto-parallel vector
field are geodesics which, for null and timelike vectors, are the paths of
freely-falling particles in general relativity. We introduce a definition of an
"auto-parallel" generalised vector field and show that it gives the analogous
statements for the classical worldvolumes of strings and branes in arbitrary
background field configurations. This appears to give a unified description of
the worldvolume equations of strings and branes, similar to the way that
generalised geometry provides a unified description of maximal supergravity
theories. We present details of the cases of string worldsheets in $O(10,10)$
generalised geometry and M2 branes restricted to the four dimensions of
$SL(5,\mathbb{R})\times\mathbb{R}^+$ generalised geometry. A key quantity is
the infinitesimal flow of the conjugate momentum along the generalised tangent
vector, which is equated to the gradient of the Hamiltonian, viewed as a
function on spacetime.
