# zbMATH — the first resource for mathematics

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