What happens if the usual geodesic equation on an n-manifold is directly modified from a source dimension 1 space (giving a path) to a dimension 2 space (giving a surface). I suspect that this gives a totally geodesic surface, but I was hoping for somewhere to discuss modifications (adding extra terms like a connection on the 2D source space) and the integrability conditions quite explicitly. 

The equations for $\gamma:\mathbb{R}^2\to\mathbb{R}^n$  are of the form
$$
\frac{\partial^2 \gamma^i}{\partial t^j\partial t^k}=D^i_{jk}(\gamma,\dot\gamma)
$$
where $D^i_{jk}$ depends on $\gamma$ and its first derivatives.


This must be well known in differential geometry, just not to me as a non-expert, so please give a reference rather than just down voting.