The cotangent space at a point on a manifold is the dual space of the tangent space at that point, consisting of all linear functionals that can act on tangent vectors. This concept is fundamental in Riemannian geometry as it relates to differential forms and provides a way to measure how functions change in different directions on the manifold. Understanding cotangent spaces allows for deeper insights into properties like gradients and differential forms, which are crucial in various applications of geometry.
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