113 research outputs found
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions
We show that most homogeneous Anosov actions of higher rank Abelian groups
are locally smoothly rigid (up to an automorphism). This result is the main
part in the proof of local smooth rigidity for two very different types of
algebraic actions of irreducible lattices in higher rank semisimple Lie groups:
(i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the
actions of cocompact lattices on Furstenberg boundaries, in particular,
projective spaces. The main new technical ingredient in the proofs is the use
of a proper "non-stationary" generalization of the classical theory of normal
forms for local contractions.Comment: 28 pages, LaTe
Positively curved manifolds with large spherical rank
Rigidity results are obtained for Riemannian -manifolds with and spherical rank at least . Conjecturally, all such
manifolds are locally isometric to a round sphere or complex projective space
with the (symmetric) Fubini--Study metric. This conjecture is verified in all
odd dimensions, for metrics on -spheres when , for Riemannian
manifolds satisfying the Raki\'c duality principle, and for K\"ahlerian
manifolds.Comment: 33 page
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