Totally geodesic immersions

Question

Let $ x: M \rightarrow \overline{M} $ be a totally geodesic immersion, where $ M $ is a $ k- $ dimensional Riemannian manifold and $ \overline{M} $ is a $ n- $ dimensional Riemannian manifold. Is it true that $ x $ is an embedding?

Thanks

Answer 1

Another example is the covering map on $S^1 $ given by $z\to z^k$ for $k\ge2$, which is "totally geodesic immersion", but not "totally geodesic embedding".

Answer 2

Not as it stands. Consider the usual irrational-slope line on a torus, which is the standard example of a 1-1 immersion thatns not an embedding. Put the flat metric on the torus and this dense curve is a geodesic.