The tangent bundle of a smooth manifold is the vector bundle whose fiber over is the tangent space .
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In differential geometry, the tangent bundle is a fundamental construction that enables the study of the properties of differentiable manifolds. It provides a way to associate a vector space (the tangent space) to each point of a manifold, facilitating the analytical treatment of curves, vector fields, and differential equations. ### Definition: For a differentiable manifold \( M \), the tangent bundle \( TM \) is defined as the collection of all tangent spaces at each point of \( M \).