It is already known that for revolution surface in $\mathbb{R}^3$, with parametric function $\langle r(s)\cos \theta,r(s)\sin \theta , z(s)\rangle$ (here $s$ is the arc length parameter for the curve $\langle r(s),z(s) \rangle$, i.e. $r'(s)^2+z'(s)^2=1$), the metric $g=ds^2+r^2(s)d\theta^2$. If we consider the curve to be a straight line, that is $r(s)=as,z(s)=\sqrt{1-a^2}s$; then the metric will have the form $g=ds^2+a^2s^2d\theta^2$, with some $|a|\leq 1$. And for $|a|>1$, this kind of metric $g=ds^2+a^2s^2d\theta^2$ will no longer be the metric of some revolution surface. So is it possible to find some surface in $\mathbb{R}^3$, s.t. the metric of this surface is $g=ds^2+a^2s^2d\theta^2$, $|a|>1$?
A metric similar to the metric of a revolution surface
1
$\begingroup$
differential-geometry
-
0Hi, I am really sorry that I made terrible typos in my question, now I have edited the problem. – 2017-01-29
1 Answers
0
This is really a local question. By letting $\varphi = c\theta$, with $c>|a|$, you reduce to the case of a (portion of a) cone, as before, with metric $ds^2+b^2s^2\,d\varphi^2$, with $0
-
0Actually, to solve this problem, it will be very helpful if you could consider the Riemann surface in complex analysis. – 2017-06-23