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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 $L^+$?

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    $\begingroup$ It is the light cone's directions, not the cone, i.e. $\mathbb{P}L^+=L^+/\mathbb{R}^+\cong S^{n-1}$. $\endgroup$ Commented Feb 28, 2021 at 15:36
  • $\begingroup$ @user10354138 That's a good answer, so it might as well be an answer, not a comment. $\endgroup$
    – Lee Mosher
    Commented Mar 1, 2021 at 0:33

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

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