The Pontryagin classes are a sequence of characteristic classes associated with real vector bundles, particularly with the tangent bundle of smooth manifolds. They provide important topological information about the manifold and are particularly used in the context of differential geometry and algebraic topology. ### Definition The Pontryagin classes \( p_i \) are typically defined for a smooth, oriented manifold \( M \) of dimension \( n \), where \( i \) ranges over integers.
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For a real vector bundle , its Pontryagin classes are the even-index Chern classes of its complexification with the displayed alternating signs. They lie in integral cohomology of degrees and obey a total-class product formula under direct sums. If a metric connection on a vector bundle has curvature matrix , its first real representative is . The Bianchi identity makes this representative closed, and Chern-Weil connection transgression makes its de Rham cohomology class independent of the connection. This normalization gives degree-four A-hat form .