I know that a surface of revolution can be parameterized in the following way: $x(s,\theta)=(r(s)\cos(\theta),r(s)\sin(\theta),s)$. In addition, the only minimal surfaces of revolution are planes and catenoids. My questions are the following:
- How to generalize surfaces of revolution to higher dimensions?
- A surface in $\mathbb{R}^3$ is minimal if and only if the trace of its shape operator is $0$. Is there a similar criterion in higher dimensions?
- What are the minimal manifolds of revolution?
Here is my attempt at answering my first question:
A surface of revolution rotates a curve $r(s)$ around a line. We can add more ways to rotate the curve around the line, for example: $x(s,\theta,\phi)=(r(s)\cos(\theta),r(s)\sin(\theta)\cos(\phi),r(s)\sin(\theta)\sin(\phi),s)$. Alternatively, we can rotate a higher dimension manifold around the line, like: $x(s,t,\theta)=(r(s,t)\cos(\theta),r(s,t)\sin(\theta),s,t)$. Combining those two ideas and generalizing to higher dimensions, we get: $x(s_1,s_2,...,s_n,\theta_1,\theta_2,...,\theta_m)=(r(s_1,...,s_n)\cos(\theta_1),r(s_1,...,s_n)\sin(\theta_1)\cos(\theta_2),...,r(s_1,...,s_n)\sin(\theta_1)\sin(\theta_2)...\sin(\theta_{m-2})\cos(\theta_{m-1}),r(s_1,...,s_n)\sin(\theta_1)\sin(\theta_2)...\sin(\theta_{m-2})\sin(\theta_{m-1}),s_1,s_2,...,s_m)$