Lie group
Let formula_1 be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map formula_2 is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism formula_3. A smooth manifold endowed with a Lie group action is also called a "G"-manifold.