Chern-Weil theorem

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Statement

Let c denote the Chern form, viz c(E,\nabla) is the Chern form for vector bundle E and connection \nabla on E. Then:

  • c_k(E,\nabla) are closed for all k and all \nabla
  • For all positive integers k, there exists a (2k-1) form T such that for all \nabla^0,\nabla^1 on E, we have:

c_k(E,\nabla^1) - c_k(E,\nabla^0) = dT(\nabla^1,\nabla^0)

Thus, c_k defines an element in H^{2k}(M,\R) independent of \nabla. This elements is termed the k^{th} Chern class (the word class because it is a cohomology class).