Basmajian identity

In mathematics , the Basmajian identity is a formula of hyperbolic geometry that relates the volume of the edge of a hyperbolic manifold to its orthospectrum.

Two geodesics orthogonal to the edge of a hyperbolic surface.

formula

Let be a compact n-dimensional hyperbolic manifold with a non-empty total geodesic boundary and the set of its geodesics orthogonal to the boundary , then we have ${\displaystyle M}$ ${\displaystyle O_{M}}$

${\displaystyle \operatorname {Vol} (\partial M)=\sum _{\gamma \in O_{M}}V_{n-1}\left(\log \left(\coth \left({\frac {1}{2}}l(\gamma )\right)\right)\right)}$,

where the length of and the volume of a ball of radius in n-1-dimensional hyperbolic space denotes. ${\displaystyle l(\gamma )}$ ${\displaystyle \gamma }$ ${\displaystyle V_{n-1}(r)}$ ${\displaystyle r}$

literature

• Basmajian, Ara: The orthogonal spectrum of a hyperbolic manifold. Amer. J. Math. 115 (1993) no. 5, 1139-1159. pdf
• Calegari, Danny: Chimneys, leopard spots and the identities of Basmajian and Bridgeman. Algebr. Geom. Topol. 10 (2010), no. 3, 1857-1863. pdf
• Vlamis, Nicholas: Moments of the length function on the boundary of a hyperbolic manifold . pdf

Web links

• Bridgeman, Martin; Tan, Ser Peow: Identities on hyperbolic manifolds
• Rivin, Igor: On Basmajian's identities, and other related stories