I was wondering a property of totally-geodesic submanifolds (planes) of the Euclidean space. Particularly, if we know a plane $\xi \subseteq \mathbb{R}^n$ that passes through a point $x \in \mathbb{R}^n$, we can get a plane through $y \in \mathbb{R}^n$ by translations. Particularly, $y - x + \xi$ would now be a plane through $y$. In fact, we can characterize all the planes that pass through $y$ in this way (i.e., they are translations of the planes through $x$).
From a differential geometry perspective, what we are essentially doing here is that we first use $\exp_x^{-1}$ to get to the origin of $\mathbb{R}^n$, and then $\exp_y$ to get to $y$. That is, given a totally-geodesic submanifold $\xi$ of $\mathbb{R}^n$ such that $x \in \xi$, then $\exp_y \circ \exp_x^{-1} (\xi)$ is a totally-geodesic submanifold of $\mathbb{R}^n$ with $y$ in it.
I was wondering if this is also possible on the sphere? One of the approaches I was thinking about was to prove that the pre-image of a totally-geodesic submanifold would be a plane in the tangent space of $x$. Even if this were possible (which I am not able to see), how would one go from the tangent space at $x$ to that at $y$? The case of $\mathbb{R}^n$ was easy since here all tangent spaces are simply translated $\mathbb{R}^n$ (i.e., the exponential map is a shift of origin). However, the same is not true for other manifolds (particularly, the sphere, in which I am interested).
Any insights into this would be highly appreciated!