On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points.
type: Item
instance of: scholarly article
title: On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points.
zbMATH Open document ID: 1065.53039
author: Gunther Uhlmann
author: Vladimir Sharafutdinov
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: This paper studies the linearized boundary rigidity problem of 2-dimensional Riemannian manifolds. A Riemannian manifold \(\left( M,g\right) \) is called boundary rigid if any Riemannian structure \(\tilde{g}\) on \(M\), that induces the same distance function on the boundary \(\partial M\) of \(M\) as \(g\), is isometric to \(g\), i.e. \(\tilde{g}=\phi^{\ast}g\) the pull-back of \(g\) via some diffeomorphism \(\phi\) of \(M\) that fixes each point of \(\partial M\). As a linearized version of the boundary rigidity, a Riemannian manifold \(\left( M,g\right) \) is called deformation boundary rigid, if the space \(Z\left( S^{2}\tau_{M}^{\prime}\right) \), representing the tangent space at \(g\) to the manifold of metrics \(\tilde{g}\) on \(M\) that induce the same distance function on \(\partial M\) as \(g\), is equal to its subspace \(P\left( S^{2}\tau _{M}^{\prime}\right) \) representing the tangent space at \(g\) to the manifold of metrics \(\tilde{g}\) isometric to \(g\) via diffeomorphisms \(\phi\) of \(M\) that fix each point of \(\partial M\), where by definition, \(Z\left( S^{2}\tau _{M}^{\prime}\right) \) consists of symmetric covariant \(2\)-tensor \(\omega\in C^{\infty}\left( S^{2}\tau_{M}^{\prime}\right) \) on \(M\) satisfying \(\int _{a}^{b}\omega\left( \gamma\left( t\right) \right) \left( \dot{\gamma }\left( t\right) ,\dot{\gamma}\left( t\right) \right) dt=0\) for any geodesic \(\gamma\) of \(\left( M,g\right) \) joining boundary points \(\gamma\left( a\right) ,\gamma\left( b\right) \in\partial M\), and \(P\left( S^{2}\tau_{M}^{\prime}\right) \) consists of \(\left( \sigma \nabla\right) \left( v\right) \), i.e. the covariant derivative \(\nabla v\in C^{\infty}\left( \tau_{M}^{\prime}\otimes\tau_{M}^{\prime}\right) \) symmetrized, for 1-forms \(v\) on \(M\) with \(v| _{\partial M}=0\). In this paper, the authors show that a compact simply connected 2-dimensional Riemannian manifold with a strictly convex boundary and with no focal points is deformation bo…
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 58J53
zbMATH DE Number: 1782663
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Riemannian surface
zbMATH Keywords: deformation boundary rigidity
zbMATH Keywords: spectral rigidity
zbMATH Keywords: Anosov manifold
zbMATH Keywords: isospectral deformation
zbMATH Keywords: focal point
reviewed by: Albert Sheu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090347526
OpenAlex ID: W1493534664
Wikidata QID: Q115203060
DOI: 10.4310/JDG/1090347526
description: scientific article; zbMATH DE number 1782663