Is it possible to define scalar function on the surface of a sphere? For instance, define a function $$ f\colon\mathbb{S}^n\to\mathbb{R}, \quad \mathbf{x}\mapsto\exp\left(-\lVert\mathbf{x}-\mathbf{x}_0\rVert^2\right). $$ In a sense that $f$ "bends" the sphere inside around $\mathbf{x}_0$.
Does this even make sense? How could the gradient of such function be calculated on $\mathbb{S}^n$?
EDIT: I'm updating this question to ask for guidelines on how to calculate the gradient of $f$ on $\mathbb{S}^n$. In fact, I have no idea where to look at, could you please help?
My goal is to define a vector field on $\mathbb{S}^n$ such, that given any $\mathbf{x}\in\mathbb{S}^n$, to be able to assign a vector $\mathbf{v}_x$ to $\mathbf{x}$ and use it in order to calculate a $\mathbf{x}^\prime=\mathbf{x}+\mathbf{v}_x$ and project it back on $\mathbb{S}^n$. I'm not sure whether the gradient of $f$ could be used for such purpose, any insight on this will be much appreciated!
Thanks!