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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

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This is the Nambu Goto action, and there is a deep theory of minimal surfaces attached to it, because there is a harmonic parametrization. –  Ron Maimon Dec 24 '11 at 0:23
    
Instead of using the area element, would it be possible to use Green's theorem? –  ben Dec 24 '11 at 4:13
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