On Deformation Boundary Rigidity and Spectral Rigidity of Riemannian Surfaces with No Focal Points

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Stability estimates for the X-ray transform of tensor fields and boundary rigidity (2004, background): “Let f 2 L2./, Igf D 0 and osf D 0 in . Then Ngf D 0 and f s Df . Part (a) immediately yields the finiteness assertion (see also [ Sh2 ]).”
Spectral rigidity and invariant distributions on Anosov surfaces (2012, background): “From the construction of these functions it is immediate that if (M, g) is free of focal points then r ≥ 0 and r− ≤ 0 (compare with [44]).”
Integral Geometry of Tensor Fields on a Class of Non-Simple Riemannian Manifolds (2006, unknown): “Some recent results include [Sh2], [Ch], [SU3], [D], [Pe], [SSU], [ ShU ].”
Boundary and lens rigidity, tensor tomography and analytic microlocal analysis (2008, background): “S-injectivity of Ig was proved in [23] for metrics with negative curvature, in [25] for metrics with small curvature and in [ 27 ] for Riemannian surfaces with no focal points.”
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions (2014, result): “For an arbitrary Anosov surface, s-injectivity of I2 was established in [41], and earlier the same result was proved in [50] if additionally the surface has no focal points.”