Given some manifold $M$ which is also equipped with some metric, for example Riemannian metric, let $$S^r(x_0)=\{x\in M| d(x_0,x)=r\},$$ where $x_0\in M$ and $d$ is the distance. Now given another point $p\in M$,
Does it make sense to talk about the submanifold $S=S^r(x_0)+p$?
Can I say that $S$ is a translation of $S^r(x_0)$ by $p$?
In general, no. For instance, consider what happens if you take the unit sphere (call it the earth), and translate the equator "by the north pole"? If you add a point on the equator to the north pole (in 3-space, where addition makes sense) you get a circle...floating in the tangent plane to the north pole, i.e., not even a subset of $S^2$.
There are some notions of translation in some manifolds, particularly ones with an ample supply of self-isometries (e.g., the torus), but ... they're not based strictly on "addition," which may not even be well-defined.