A manifold is a mathematical space that locally resembles Euclidean space near each point, meaning that in small enough neighborhoods, it looks like standard flat space. Manifolds can have various dimensions and can be either smooth or have certain structures imposed on them, which allows for the study of complex shapes and spaces in a more manageable way. This concept is crucial in understanding the geometric structures that arise in the study of Lie groups and their corresponding Lie algebras.
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