## Duke Mathematical Journal

### A sum formula for a pair of closed geodesics on a hyperbolic surface

Nigel J. E. Pitt

#### Abstract

We consider an arbitrary pair of closed geodesics and the corresponding period integrals for the eigenfunctions of the Laplacian on a compact hyperbolic surface. A summation formula that relates geometric information about the geodesics (namely, the angles of intersection and lengths of common perpendiculars between them) to the period integrals is proved. As a corollary, an asymptotic is obtained for the second moment of the period integrals for a fixed geodesic as an average over the eigenvalue with an error term that can be interpreted in terms of the geometric data

#### Article information

Source
Duke Math. J., Volume 143, Number 3 (2008), 407-435.

Dates
First available in Project Euclid: 3 June 2008

https://projecteuclid.org/euclid.dmj/1212500462

Digital Object Identifier
doi:10.1215/00127094-2008-024

Mathematical Reviews number (MathSciNet)
MR2423758

Zentralblatt MATH identifier
1195.11073

#### Citation

Pitt, Nigel J. E. A sum formula for a pair of closed geodesics on a hyperbolic surface. Duke Math. J. 143 (2008), no. 3, 407--435. doi:10.1215/00127094-2008-024. https://projecteuclid.org/euclid.dmj/1212500462

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