TITLE: Killing vector fields on sphere
QUESTION [8 upvotes]: Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.
$$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$
Is $u$ necessarily constant?

REPLY [6 votes]: The answer is "no".
Choose a basis $V_1, V_2, V_3$ of Killing fields.
Note that
$$\int\limits_{\mathbb S^2} V_iu\cdot x_j\cdot d\,\mathrm{area}
=
-\int\limits_{\mathbb S^2} u\cdot V_ix_j\cdot d\,\mathrm{area}$$
Threfore you can take any $u$ which is orthogonal to each of 9 functions $s_{i,j}=V_ix_j$.