Sep 15, 2023 · $\begingroup$ I just wonder if the metric of the hyperboloid is that of a 2-sphere, but with the sine replaced by a hyperbolic sine. $\endgroup$ – Il Guercio Commented Sep 15, 2023 at 7:50

Sep 19, 2023 · $\begingroup$ @LeeMosher OP has given the pullback of the Euclidean metric to the hyperboloid in (global) local coordinates on the hyperboloid. We assume (perhaps wrongly) that OP is looking for the induced metric on the hyperboloid in $\Bbb R^5$, in terms of coordinates on the hyperboloid. $\endgroup$ –

Jul 28, 2023 · $\begingroup$ The easiest way is to show that the velocity vector of the curve in Minkowski space is always orthogonal to the hyperboloid. $\endgroup$ – Deane Commented Jul 28, 2023 at 15:56

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a two-s...

I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived. I would appreciate it if either someone could explain to me how such a parameterization is derived or recommend a reference.

Oct 30, 2020 · The result is a hyperboloid of two sheets contained within a double cone. The equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1, \tag2$$ on the other hand, is a hyperboloid of one sheet.

What user69810 is suggesting is to transform the equation for the hyperboloid into cylindrical coordinates and carry out the volume integration in the three coordinate variables. (This sounds worse than it actually is for this symmetrical figure.)

Find the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$.

Jul 15, 2016 · This is however a lot of work and for me only possible in some very nice cases where it's particularly easy, not every time. Is there any criterion that immediately tells me if it is one-sheeted or two-sheeted, like in $\mathbb{R}^2$, where you can immediately tell if the solution set of a quadric is an ellipse or a hyperboloid? Any help ...

Dec 4, 2018 · The upshot here is that the pairs (hyperboloid, euclidean metric) and (hyperboloid, Minkowski metric) are distinct geometric objects. The way to measure lengths and areas is not the same in these two "worlds". The first one has non-constant positive curvature, and the second one has constant negative curvature $-1$.