My question is why for an orthogonal action of G on $S^n$ the fixed point set is again sphere $S^r$
Orthogonal action of G means G has orthogonal representation that is there exist a homeomorphism of G into O(n).
Fixed point set of G on X is the set of those x in X such that g(x)=x for all g in G
For action of O(3) on $S^2$ the fixed point set is $S^0$ the two point set.but i don't know how to prove it generally . Thanks in advance