Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.

Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.

Question 1: Is F the identity map?

If the answer is negative: Is $F$ an isometry?