Abstract
The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperimetric inequalities etc.). We will restrict our attention to those results that have a direct connection to the boundary rigidity problem (see Section 2) or the conjugacy rigidity problem (see Section 4). Even with that restriction the connections are numerous and the author was forced to select the topics covered here in accordance with rather subjective criteria. A few of the topics not covered are mentioned in Section 11 but there are others.KeywordsRiemannian ManifoldSymmetric SpaceNegative CurvatureConjugate PointClosed GeodesicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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