In
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
, McShane's identity for a
once punctured
In topology, puncturing a manifold is removing a finite set of points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point, it becomes twice ...
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
with a complete, finite-volume hyperbolic structure is given by
:
where
* the sum is over all simple closed
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
γ on the torus; and
* ''ℓ''(''γ'') denotes the hyperbolic length of ''γ''.
This identity was generalized by
Maryam Mirzakhani
Maryam Mirzakhani ( fa, مریم میرزاخانی, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ...
on her PhD thesis
References
{{Reflist
*''Necessary and Sufficient Conditions for McShane's Identity and Variations'' Ser Peow Tan, Yan Loi Wong, and Ying Zhang eprint arXiv:math/041118
*McShane, G. Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632.
Geometric topology