About: Lie–Palais theorem

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In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary, in other words the vector fields on the boundary must be tangent to the boundary. Palais proved it as a global form of an earlier local theorem due to Sophus Lie. The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds.

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rdf:type	yago:WikicatLieAlgebras yago:WikicatLieGroups yago:WikicatTheoremsInDifferentialGeometry yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Communication100033020 yago:Content105809192 yago:Discipline105996646 yago:Group100031264 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:Message106598915 yago:Proposition106750804 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 yago:Science105999797 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Lie–Palais theorem (en)
rdfs:comment	In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary, in other words the vector fields on the boundary must be tangent to the boundary. Palais proved it as a global form of an earlier local theorem due to Sophus Lie. The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds. (en)
dcterms:subject	Lie algebras Theorems in differential geometry
Wikipage page ID	37647580 (xsd:integer)
Wikipage revision ID	971172877 (xsd:integer)
Link from a Wikipage to another Wikipage	Vector field Lie group Lie algebras Lie algebra Group action (mathematics) Theorems in differential geometry Differentiable manifold Differential geometry Sophus Lie Unit interval Compact manifold
sameAs	Lie–Palais theorem Lie–Palais theorem Lie–Palais theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Short_description dbt:Harvs
has abstract	In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary, in other words the vector fields on the boundary must be tangent to the boundary. Palais proved it as a global form of an earlier local theorem due to Sophus Lie. The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds. Without the assumption that the Lie algebra is finite dimensional the result can be false. , p. 1048) gives the following example due to Omori: the Lie algebra is all vector fields f(x,y)∂/∂x + g(x,y)∂/∂y acting on the torus R2/Z2 such that g(x, y) = 0 for 0 ≤ x ≤ 1/2. This Lie algebra is not the Lie algebra of any group. gives an infinite dimensional generalization of the Lie–Palais theorem for Banach–Lie algebras with finite-dimensional center. (en)
prov:wasDerivedFrom	wikipedia-en:Lie–Palais_theorem?oldid=971172877&ns=0
page length (characters) of wiki page	2425 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Lie–Palais_theorem
is Link from a Wikipage to another Wikipage of	Lie group action Palais theorem Richard Palais List of theorems List of things named after Sophus Lie Lie-Palais theorem
is Wikipage redirect of	Lie-Palais theorem
is Wikipage disambiguates of	Palais theorem
is foaf:primaryTopic of	wikipedia-en:Lie–Palais_theorem

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