For closed orientable hyperbolic surfaces of curvature , the spectrum of the Laplace-Beltrami operator, including multiplicities, determines the unmarked length spectrum of closed geodesics, and conversely. Count either oriented geodesics throughout or unoriented ones throughout. Primitive lengths with multiplicities determine the lengths of all iterates, and these determine the primitive lengths recursively.[ref]
Huber's theorem for hyperbolic orbisurfaces by Emily B. Dryden and Alexander Strohmaier. Section 1, classical Huber theorem and oriented length-spectrum convention
The Selberg trace formula connects the two lists: the spectral side sums functions of eigenvalues, while the geometric side sums explicit contributions from closed-geodesic lengths. Equality of either complete list forces equality of the other. The theorem concerns these spectral data; distinct surfaces can still have equal spectra.
For nonorientable hyperbolic surfaces, the trace contributions distinguish orientation-preserving and orientation-reversing geodesics. Their separate counts, rather than an unspecified undifferentiated convention, belong in the extended statement.[ref]
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