In the derivation of the geodesic, one starts with the integral of the line element (arclength):
$$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$
The integrand is then substituted into the Euler-Lagrange equation, which simplifies to the geodesic equation.
My question is would it be possible to start with the area element instead of the line element, and thereby arrive at a surface area geodesic equation, which I guess would be an equation for the surface area of a given patch of manifold?
Thanks
