Divergence free vector field on spheres

Question

Let $S^{n-1}$ be the unitary sphere in $\mathbb{R}^n$. Let $Y\in C^\infty(\mathbb{R}^n;\mathbb{R}^n)$ be a zero-homogeneous vector field, that is $$ Y(x)=Y\left(\frac{x}{|x|}\right)\quad\mbox{in }\mathbb{R}^n, $$ and consider the decomposition $$ Y=\left(Y\cdot \frac{x}{|x|}\right)\frac{x}{|x|} + Y_T, $$ where $Y_T$ is the tangential component of $Y$ to $S^{n-1}$. Now, suppose that $$ \mathrm{div}_{S^{n-1}}(Y_T)=0 \quad\mbox{in }S^{n-1} $$ where $\mathrm{div}_{S^{n-1}}$ is the tangential divergence on $S^{n-1}$: can we say that $Y_T=\nabla_{S^{n-1}}f$ for some function $f\colon S^{n-1} \to \mathbb{R}$ or more information like $Y_T\equiv 0$ or for the radial component $(Y\cdot x/|x|)x/|x|$? Probably this is a trivial question, but I cannot find a simple reference. Moreover, if this is true, is it possible to lighten the regularity assumptions on $Y$?