A smooth action of the compact abelian Lie group . Commuting vector fields with a common periodic normalization give such an action. In action-angle variables, the torus acts by translation of the angles while leaving the actions fixed.
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A **Torus action** refers to the action of a torus (typically a compact, connected Lie group isomorphic to the product of several circles, denoted as \(T^n\), where \(n\) is the number of circles) on some space, often a manifold. This concept arises in various areas of mathematics, including differential geometry, algebraic geometry, and symplectic geometry.