### Abstract

Let X be a compact Riemannian manifold with conic singularities, i.e. a Riemannian manifold whose metric has a conic degeneracy at the boundary. Let Δ be the Friedrichs extension of the Laplace-Beltrami operator on X. There are two natural ways to define geodesics passing through the boundary: as "diffractive" geodesics which may emanate from ∂X in any direction, or as "geometric" geodesics which must enter and leave ∂X at points which are connected by a geodesic of length π in ∂X. Let DIFF = {0}∪{±lengths of closed diffractive geodesics} and GEOM = {0} ∪ {±lengths of closed geometric geodesics}. We show that Tr cos t√Δ ∈ ^{-n-0} (ℝ) ∩ C^{-1-0} (ℝ\GEOM) ∩ C^{∞} (ℝ\DIFF). This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case X = S^{1}.

Original language | English (US) |
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Pages (from-to) | 813-828 |

Number of pages | 16 |

Journal | Mathematical Research Letters |

Volume | 9 |

Issue number | 5-6 |

State | Published - Sep 1 2002 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematical Research Letters*,

*9*(5-6), 813-828.