RBF kernel's explicit feature map, which isn't unique, (as given in Slide 11 of this link) is (it gives the mapping only for for 1D case):
$$\phi(x)=e^{-\gamma x^2}\left[1,\sqrt{\frac{2\gamma}{1!}}x,\sqrt{\frac{(2\gamma)^2}{2!}}x^2,...\right]$$
If we calculate the Euclidean norm of this vector, we'd have:
$$\Vert\phi(x)\Vert^2=e^{-2\gamma x^2}\sum_{i=0}^\infty \frac{(2\gamma)^i}{i!}x^{2i}=e^{-2\gamma x^2}\sum_{i=0}^\infty\frac{(2\gamma x^2)^{i}}{i!}=e^{-2\gamma x^2}e^{2\gamma x^2}=1$$
So, the norm of the vectors are $1$ ($\Vert\phi(x)\Vert^2=1\rightarrow \Vert\phi(x)\Vert=1)$, which means the mapping is onto surface of unit hypersphere in infinite dimensions. I'd gladly try to prove this for larger dimensions if I can find an explicit expression.