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Local boundary rigidity of a compact Riemannian manifold with curvature bounded above. (English) Zbl 0958.53027

The authors study the following rigidity problem: to which extent is a Riemannian metric on a compact manifold \((M,g)\) with boundary \(\partial M\) determined from the distances between boundary points. Such a Riemannian manifold \((M,g)\) is called boundary rigid if for any other metric \(g'\) on \(M\) with the same boundary distance function there exists a diffeomorphism \(\varphi: M\to M\) which is the identity on \(\partial M\) such that \(g'= \varphi^*g\).
The authors consider dissipative manifolds, i.e., compact Riemannian manifolds \((M,g)\) with convex boundary such that for every tangent vector \(v\in TM\) the geodesic \(c_v\) with \(\dot c_v(0)= v\) is defined on a bounded interval \([\tau_-(v), \tau_+(v)]\). They prove under an additional curvature assumption which is always satisfied in the case of nonpositive curvature or sufficiently small convex domains that dissipative manifolds are locally boundary rigid.

MSC:

53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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