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Restriction of the geodesic distance to an arc and rigidity. (Restriction de la distance géodésique à un arc et rigidité.) (French) Zbl 0821.53038

Let \(g_ 0\), \(g_ 1\) be two metrics of class \(C^ \infty\) (resp. analytic) defined on an open set \(\theta\) of \(\mathbb{R}^ 2\). Let \(AB\) be an arc of a curve of class \(C^ 1\) imbedded in \(\theta\) starting at \(A\) and ending in \(B\). We assume that for \(i \in \{0,1\}\) and for all pairs \(u,v \in AB\), there exists a unique geodesic \(\gamma^ i_{uv}\) relative to the metric \(g_ i\), \(\gamma^ i_{uv} : [0,1] \to \theta\) so that \(\gamma^ i_{uv}(0) = u\), \(\gamma^ i_{uv}(1) = v\), \(\gamma^ i_{uv}(]0,1[) \cap AB = \varnothing\) and \(\text{dist}_ 0(u,v) = \text{dist}_ 1 (u,v)\), where \(\text{dist}_ i\) denotes the distance induced by \(g_ i\). Then there exists a diffeomorphism of class \(C^ \infty\) (resp. analytic) \(\delta : \theta_ 0' \to \theta_ 1'\) where \(\theta_ 0'\), \(\theta_ 1'\) denote open sets such that \(AB \subset \theta_ 0' \subset \theta\), \(AB \subset \theta_ 1' \subset \theta\) which is the identity on \(AB\), and that the metrics \(\delta^* g_ 1\) and \(g_ 0\) have the same jet of order infinite in every point of \(AB\) (resp. \(\delta^* g_ 1 = g_ 0\), so that \(g_ 0\) and \(g_ 1\) are isometric). In short, one shows how only the local restriction of the geodesic distance to a convex curve, allows to identify the metric structure of a surface, under certain analytical assumptions.

MSC:

53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)

Keywords:

rigidity; geodesic
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References:

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