Editing Chern class (section)

=== Basic idea and motivation ===

Chern classes are [[characteristic class]]es. They are [[topological invariant]]s associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.

In topology, differential geometry, and algebraic geometry, it is often important to count how many [[linearly independent]] sections a vector bundle has. The Chern classes offer some information about this through, for instance, the [[Riemann–Roch theorem]] and the [[Atiyah–Singer index theorem]].

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the [[curvature form]].