What is the boundary at infinity in the hyperboloid model of the hyperbolic plane?

Question

The boundary of the upper half-plane $\mathbb{H}$ model of the hyperbolic plane is the extended real line $\overline{\mathbb{R}}= \mathbb{R} \cup \{ \infty \}$.

What is the boundary of the hyperboloid model of the hyperbolic plane? Is it the positive light cone $L^+$?

It is the light cone's directions, not the cone, i.e. $\mathbb{P}L^+=L^+/\mathbb{R}^+\cong S^{n-1}$. — user10354138, Feb 28 '21 at 15:36
@user10354138 That's a good answer, so it might as well be an answer, not a comment. — Lee Mosher, Mar 01 '21 at 00:33

score 0 · Answer 1 · answered Mar 06 '21 at 03:08

0

The boundary is not the light cone $L_+$ itself, but the set of directions of the light cone, or equivalently the projectivization $L_+ / \mathbb R_+ \cong S^{n-1}$.

answered Mar 06 '21 at 03:08

Lee Mosher

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What is the boundary at infinity in the hyperboloid model of the hyperbolic plane?

1 Answers1