Recent progress on the boundary rigidity problem.
Riemannian foliations with parallel or harmonic basic forms
In this paper, we consider a Riemannian foliation that admits a nontrivial parallel or harmonic basic form. We estimate the norm of the O’Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.
Rigid Two-Step Nilpotent Lie Groups relative to Multicontact Structures
Rigidity and cohomology of hyperbolic manifolds
When is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.
Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces
Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.
Rigidity in non-negative curvature
Rigidity of minimal hypersurfaces of spheres with constant Ricci curvature.
Rigidity of symmetric spaces
Rigidity of the stable norm on tori.
Rigidity Theorems of Hypersurfaces in a Sphere