On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points.
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 boundary rigid. Let us call a closed (boundaryless) Riemannian manifold \(\left( M,g\right) \) deformation rigid if the above definition for deformation boundary rigidity is satisfied with \(\partial M=\emptyset\) and with geodesics \(\gamma\) between boundary points replaced by closed geodesics of \(\left( M,g\right) \). It is shown that any 2-dimensional Anosov manifold, i.e. any 2-dimensional closed Riemannian manifold with geodesic flow of Anosov type, with no focal points is deformation rigid. Furthermore a closed Riemannian manifold \(\left( M,g\right) \) is called spectrally rigid if any isospectral deformation of \(g\), i.e. a smooth one-parameter family of metrics \(g^{\tau}\) with \(g^{0}=g\) and with the spectrum of the Laplace-Beltrami operator \(\Delta^{\tau}\) of \(g^{\tau}\) independent of \(\tau\), is a trivial deformation, i.e. \(g^{\tau}=\left( \phi^{\tau}\right) ^{\ast}g\) for a family of diffeomorphisms \(\phi^{\tau}\) of \(M\). As a corollary, it is derived that any 2-dimensional Anosov manifold with no focal points is spectrally rigid.
scientific article; zbMATH DE number 1782663