Le spectre marqué des longueurs des surfaces à courbure négative. (The spectrum marked by lengths of surfaces with negative curvature). (French) Zbl 0699.58018

Otal, Jean-Pierre

Ann. Math. (2) 131, No. 1, 151-162 (1990).

Let S be an orientable closed surface with the genus \(\geq 2\). It is known that for a Riemannian metric m with negative curvature any nontrivial conjugate class \(<\gamma >\) can be represented by a unique geodesic of m. Consider \(\ell (\gamma)\) the length of such geodesic. If \({\mathcal C}\) represents the set of the conjugate classes of the fundamental group \(\pi_ 1(S)\) one defines the length’s spectrum of the metric m by the element \((\ell (\gamma))_{\gamma \in {\mathcal C}}\) of the direct product \({\mathbb{R}}^{{\mathcal C}}\). Considering \({\mathcal M}(S)\) the space of the metrics m with negative curvature, one obtains the function \({\mathcal L}: {\mathcal M}(S)\to {\mathbb{R}}^{{\mathcal C}}\) defined by \(m\mapsto (\ell (\gamma))_{\gamma \in {\mathcal C}}.\)
The main purpose of this paper is the proof of the following result: the function \({\mathcal L}\) is one-to-one, i.e. the metrics with negative curvature on S are determined, via an isometry, by their lengths spectrum.
Remark that this result was conjectured by K. Burns and A. Katok [Ergodic Theory Dyn. Syst. 5, 307-317 (1985; Zbl 0572.58019)] and the problem of injectivity of the function \({\mathcal L}\) appears in V. Guillemin and D. Kazhdan [Topology 19, 301-312 (1980; Zbl 0465.58027)] and A. Katok [Ergodic Theory Dyn. Syst. 8, 139-152 (1988; Zbl 0668.58042)].

Reviewer: D.Andrica

MSC:

58C40	Spectral theory; eigenvalue problems on manifolds
58D17	Manifolds of metrics (especially Riemannian)
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
53C20	Global Riemannian geometry, including pinching

Keywords:

metric with negative curvature; spectrum marked by lengths; Riemannian metric; geodesic; isometry