The tangent space at a point on a manifold is a vector space that consists of all possible tangent vectors at that point. It serves as a way to capture the local structure of the manifold and allows for the analysis of curves and surfaces in its vicinity. Understanding the tangent space is essential for discussing concepts like parallel transport, which involves moving vectors along curves on the manifold, and it is also crucial when differentiating between covariant and contravariant vectors.
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