A smooth manifold is a topological space that locally resembles Euclidean space and has a differentiable structure, allowing for calculus to be performed on it. This means that around every point, there is a neighborhood that can be mapped to an open set in Euclidean space, and transition maps between these neighborhoods are smooth functions. Smooth manifolds are essential in understanding the geometry and topology of Lie groups since many Lie groups can be modeled as smooth manifolds.
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