The Maurer-Cartan connection is a specific type of connection defined on the tangent bundle of a Lie group, which encodes how to differentiate along curves in the group. It provides a way to relate the algebraic structure of the Lie group to the geometric structure of its tangent space, allowing us to understand how elements and their infinitesimal variations behave under group operations. This connection is essential for studying properties such as curvature and holonomy in the context of differential geometry.
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