The tangent bundle of a smooth manifold is the vector bundle whose fiber over is the tangent space .
For the real tautological bundle over Real projective space, the tangent space at a line is . Adding gives , using a metric to identify a real line with its dual. Hence its total Stiefel–Whitney class is , with .
A coordinate chart trivializes the tangent bundle by expressing in the coordinate basis. For , the chain rule gives the displayed transition. Smoothness and fibrewise linear invertibility of the Jacobian matrix construct the natural smooth vector bundle structure. The chain rule also supplies the transition cocycle.
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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 \( M \).