Consider (one sheet of) a hyperboloid of two sheets embedded in Euclidean $\mathbb{R}^3$. There are at least two natural geometric structures on this surface:

  1. The intrinsic geometry inherited from the embedding in $\mathbb{R}^3$.

  2. The hyperboloid model of the hyperbolic plane. The hyperboloid sits in a unique cone in $\mathbb{R}^3$ and the geodesics of this model are the intersections of the hyperboloid with Euclidean planes through the apex of the cone.

Question: Given two intersecting geodesics in the hyperboloid model, the angle between them can be measured using either of the metrics 1 or 2. Do these two measurements coincide?

Bonus Question: What do the "circles" of metric 2 look like in terms of metric 1?

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    $\begingroup$ Remember that you need the Lorentz metric on $\Bbb R^3$ to induce a constant negative curvature metric on the hyperboloid of two sheets. $\endgroup$ Feb 12 '17 at 17:47
  • $\begingroup$ Fixed the typo. $\endgroup$ Feb 12 '17 at 23:05
  • $\begingroup$ @DrewArmstrong Drew, I added a comment concerning the circles in the hyperboloid model. I didn't see the bonus question earlier. $\endgroup$ Feb 15 '17 at 20:42

If you are asking about 1) the usual Euclidean metric on $\mathbb{R}^3$ restricted to the (upper sheet of the) hyperboloid versus 2) the constant negative curvature metric (i.e. the hyperbolic metric) on that hyperboloid, which is obtained by restricting the Minkowski metric of $\mathbb{R}^{2,1}$ on it, then the answer is no, the hyperboloid model is not conformal. The angles measured between two hyperbolic geodesics with respect to 1) are in general different from angles measured with respect to 2). The only point of conformality (where the two angle measurements always coincide) is the apex of the sheet. Occasionally, there could be some coincidence for some very special cases, but generically the angles are different.

I forgot the bonus question: the circles in the hyperboloid model are intersection of the hyperboloid sheet with space-like planes, not passing through the origin. In particular, if you choose a point on the hyperboloid, which means you have a Minkowski-unit vector from the origin to the chosen point, you can take the space-like plane Minkowski-orthogonal to that vector and passing through the point. The latter plane is tangent to the hyperboloid. By taking planes, parallel to that one and shifted above it so that they intersect the hyperboloid, you get the family of concentric circles centered at the chosen point. Similarly, horocycles are the planes that are paralle to a light-like vector (null-vector) and offset from the origin, so that when they intersect the hyperboloid, a parabola is formed.


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