I have a curve in $\mathbb{R^3}$ as $z= f(x)$ in the $xz$- plane and we let $S$ be the surface generated from this curve in the space by revolving it about the $z$- axis. Now, I'm asked to find it's curvature. Also, it's provided that the metric for this surface is the induced standard Euclidean Riemannian-metric.
Now, to find this, it's easy to see that the Riemannian metric in this case is $ds^2= dx^2+ dy^2+dz^2$ but to avoid complications in the computations later I let it be spherical or cylindrical version of this?
Now, I'll have some expressions for metric coefficients as $g_ij$, which will then give me Christoffel symbols from there I can find curvature tensor.
Another approach I was thinking of was to find the expression for the surface here in $x,y,z$ and then use sectional curvature of a surface = Gaussian curvature of the surface for which we have a nice formula.
So, is this a correct way to go on about finding solutions of this problem? It'd be great if someone could cross-check it for me or point out a flaw in my approach.
Thank you.
Be careful, ${\rm d}x^2+{\rm d}y^2 + {\rm d}z^2$ is not the Riemannian metric in $S$. This is the Riemannian metric of $\Bbb R^3$, and the metric in $S$ is the pull-back of that via the inclusion $S\hookrightarrow \Bbb R^3$. Let's find natural coordinates for $S$, by noting that its points are obtained by rotating $z = f(x)$ around the $z$ axis: $$\begin{pmatrix} \cos \theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix} x \\ 0 \\ f(x)\end{pmatrix} = \begin{pmatrix} x\cos \theta \\ x \sin \theta \\ f(x)\end{pmatrix}.$$So we can compute the pull-back of the Euclidean metric as $${\rm d}(x\cos\theta)^2 + {\rm d}(x\sin\theta)^2 + {\rm d}f^2 = (1+f'(x)^2){\rm d}x^2+x^2{\rm d}\theta^2.$$Then you can use this expression to compute Christoffel symbols and Riemann tensor components. Although an easier strategy is to just find an unit normal vector to $S$, and compute the shape operator, just like books on curves and surfaces do when they're trying to make the material introductory and avoid the language of manifolds, etc. (see e.g., Ted Shifrin's notes).