Lie–Palais theorem

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. 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. 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.