Consider the real valued function $f(x,y,z) = (2-(x^2+y^2)^{\frac{1}{2}})^2 + z^2$ on $R^3-{(0,0,z)}$. Show that 1 is a regular value of f. Identify the manifold $M=f^{-1}(1)$.
I showed that 1 is regular value of f by proving that $f_*$ is onto for all points $(a,b,c)$ such that $f(a,b,c)=1$. Hence $M=f^{-1}(1)$ is an embedded submanifold of dimension 2 on $R^3$. Can you help me with the second part? Should I give a description of such points or do I need to include charts or functional structures? Thank you!