Let $\mathbb{S}^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\colon\lVert\mathbf{x}\rVert^2=1\}$ be the unit $n$-sphere and $\mathbf{x},\mathbf{x}^\prime\in\mathbb{S}^n$ two points on it.
I would like to investigate if it possible to define a unique $n$-dimensional vector, living in a tangent space (like the ones shown in the figure below), so as "traveling" across the dashed geodesic is possible without defining a new "direction" at any given point of $\mathbb{S}^n$ (e.g., on $\mathbf{x}$ and $\mathbf{x}^\prime$).
As an analogy, I'm thinking of the Euclidean $n$-dimensional space, where a given vector is "enough" to define a unique direction. Roughly speaking, we can think that if we let an object move forward/backward on a given direction (i.e., defined by a single vector), then no further information (i.e., a different vector) is needed at any given point of $\mathbb{R}^n$.
Similarly, my intuition says that if a single direction (in terms of a vector that belongs to the tangent space at a given point of $\mathbb{S}^n$) is given to a point of $\mathbb{S}^n$, then we can travel along a geodesic and arrive at the same point, without the need to change "direction", as long as we stay on the surface of the manifold. Thus, I wonder if what I mentioned about the Euclidean space is possible in this non-Euclidean manifold (or even in other non-Euclidean manifolds, like n-torus etc), and what would be the minima conditions for that. For instance, in the Euclidean space example, we need to define an $n$-dimensional vector.
In other words: can the geodesic on $\mathbb{S}^n$ shown above be defined uniquely by a vector?