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?