differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the Lie–Palais theorem states that an

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of a finite-dimensional

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

on a

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

compact manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is ...

can be lifted to an action of a finite-dimensional

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

. 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. proved it as a global form of an earlier local theorem due to

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sophu ...

. The example of the vector field ''d''/''dx'' on the open

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

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. 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.

References

* Reprinted in collected works volume 5. * * {{DEFAULTSORT:Lie-Palais theorem Lie algebras Theorems in differential geometry