Surfaces extending modified geodesic paths

Question

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 $$ \frac{\partial^2 \gamma^i}{\partial t^j\partial t^k}=\Xi^p_{jk} \frac{\partial \gamma^i}{\partial t^p} -(\Gamma^i_{pq}\circ\gamma) \frac{\partial \gamma^p}{\partial t^j} \frac{\partial \gamma^q}{\partial t^k} $$ where $\Gamma$ is the Christoffel symbols on $\mathbb{R}^n$ and $\Xi$ is the Christoffel symbols on $\mathbb{R}^2$. Possibly add some tensor evaluated at $\gamma$ as well...

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.

Answer 1

Assume that $\gamma\colon\Sigma\to M$ is a map from a surface to an $n$-manifold, both equipped with Riemannian metrics and theire Levi-Civita connections. Then there exists a connection $\nabla^\gamma$ along $\gamma$ induced by $\nabla^M$. Let me rewrite your equation as $\nabla^\gamma_X(d\gamma\circ Y)=d\gamma(\nabla^\Sigma_XY)$. Here $X$, $Y$ are vector fields on $\Sigma$, and the equation is in vector fields along $\gamma$.

Assume that $\gamma$ satisfies this equation, let $c\colon I\to\Sigma$ be a geodesic, and let $X$ extend the vector field $\dot c$. Then $\nabla^\Sigma_XX$ vanishes along $c$, so the right hand side vanishes along $c$. The left hand side along $c$ can be read as the geodesic equation for $\gamma\circ c$. Hence $\gamma$ maps geodesics to geodesics if our equation holds.

So $\gamma$ is a projective embedding with totally geodesic image ("projective" means "maps geodesics to geodesics". Here, we also map constant speed parametrisations to constant speed parametrisations, which is close (but not equivalent) to being isometric up to scaling). As totally geodesic submanifolds do not exist in generic Riemannian manifolds $(M,g)$, you are in very special situation if your equation holds.