Surface of Revolution
$\gamma(u)=(f(u),0,g(u))$ and
$\sigma(u,v)=(f(u)\cos v, f(u)\sin v, g(u))$
Fist of all, I calculated the first fundametal form for surface of revolution.
And I obtained that $$d\theta^2+\cos^2\theta \ d\varphi^2$$
An then, I calculated the second fundametal form for surface of revalution.
I obtained that $$(\dot f(u)\ddot g(u)-\ddot f(u)\dot g(u))du^2 +f(u)\dot g(u) dv^2$$
When I take $u=\theta$ $v=\varphi$ $f(\theta)=\cos \theta$ and $g(\theta)=\sin \theta$
I get the result $$d\theta^2+\cos^2\theta \ d\varphi^2$$
That's, its 1st fundametal form and second fundametal form are the same.
Why? How does there exist a relation between them? Please explain it. Thank you.
In general, the first and second fundamental forms of a surface are different objects. Let $r=r(u,v)$ be a parametrization of a given surface and let $n=n(u,v)$ denote the normal vector. We use the notation
$$I:=Edu^2 + 2Fdudv + Gdv^2$$ where $E = r_u\cdot r_u; F = r_u\cdot r_v;G = r_v\cdot r_v$.
for the first fundamental form, and
$$II:=Ldu^2 + 2Mdudv + Ndv^2$$ where $L = r_{uu}\cdot n$; $M = r_{uv}\cdot n$; $N = r_{vv}\cdot n$ for the second one.
If we consider the unit sphere, i.e. $r(u,v)=(\sin u\sin v, \sin u\cos v,\cos u)$, then
$$I=II$$
as the normal vector satisfies $n(u,v)=r(u,v)$ (this is evident geometrically) and
$$L=E, $$ $$M=F, $$ $$N=G; $$
this follows from the very definitions of the coefficients themselves.
Remark: if the sphere has radius $a>0$ then, in general, $$aII=I$$.