Deformation of Locally Free Actions and Leafwise Cohomology

Asaoka, Masayuki

doi:10.1007/978-3-0348-0871-2_1

Cited by 5 publications

(7 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There exists a unique action ρ ′ of N on M whose orbit foliation is F and canonical 1-form is η. See [1]. By parameter rigidity ρ ′ is conjugate to ρ.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Parameter rigid actions of simply connected nilpotent Lie groups

Maruhashi

2012

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We show that for a locally free C ∞ -action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [8] concerning the Heisenberg groups.

show abstract

“…There exists a unique action ρ ′ of N on M whose orbit foliation is F and canonical 1-form is η. See [1]. By parameter rigidity ρ ′ is conjugate to ρ.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Parameter rigid actions of simply connected nilpotent Lie groups

Maruhashi

2012

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…Proof. Under the assumptions of this corollary, (2) in Lemma 7 follows from the case λ = 0 in Corollary 11 below, and (3) in Lemma 7 follows from Theorem 8. This implies parameter rigidity of the action.…”

Section: H Maruhashimentioning

confidence: 96%

“…Note that x * ω 0 is the canonical 1-form of ρ. By Proposition 1.4.4 of Asaoka [2], there exist ∈ Aut(S) and a C ∞ map P :…”

Section: Vanishing Of H 0 -Proof Of Theorem 28mentioning

confidence: 98%

Section: Vanishing Of H 0 -Proof Of Theorem 28mentioning

confidence: 99%

“…where denotes the left Maurer-Cartan form of S. (In [2], is referred to as an endomorphism, but it is the same that appears in the definition of parameter equivalence which we saw in §2, so can be taken as an automorphism. It is easy to see P * is equivalent to the expression P −1 d F P in [2]. There is a small difference between our definition of parameter equivalence and the one in [2], since in [2] the map F is assumed to be homotopic to the identity through diffeomorphisms.…”

Section: Vanishing Of H 0 -Proof Of Theorem 28mentioning

confidence: 99%

See 2 more Smart Citations

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Maruhashi

2020

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity. First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type. Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

show abstract

Local rigidity problem of smooth group actions

Asaoka

2017

Sugaku Expositions

View full text Add to dashboard Cite

Deformation of Locally Free Actions and Leafwise Cohomology

Cited by 5 publications

References 41 publications

Parameter rigid actions of simply connected nilpotent Lie groups

Parameter rigid actions of simply connected nilpotent Lie groups

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Local rigidity problem of smooth group actions

Contact Info

Product

Resources

About