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Classical worldvolumes as generalised geodesics

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.