Non-trivial vector field with divergence $0$ on $\mathbb{S}^{2}$.

Question

I am looking for a non-trivial vector field $X$ on the sphere $\mathbb{S}^{2}\subseteq\mathbb{R}^{3}$ with the induced metric such that its divergence is $0$. I recall here the expresion of the divergence of a vector field in a chart: $$ X=\sum_{i=1}^{n}X_{i}\frac{\partial}{\partial x_{i}}\Rightarrow \mathrm{Div}X=\frac{1}{\sqrt{|g|}}\sum_{i=1}^{n}\frac{\partial(X_{i}\sqrt{|g|})}{\partial x_{i}}, $$ where $g$ is the determinant of the matrix of the metric $(g_{i,j})_{1\leq i,j\leq n}$. Does it exist?

Answer 1

One example would be to simply rotate the sphere:

$$ X(x,y,z) = (-y,x,0) $$

(This considers the sphere to be embedded in $\mathbb R^3$, and has values in the induced representation of the tangent space at each point. I'll leave it to you to represent it in a two-dimensional chart of your choice).