Chern Classes and Numbers

Chern Classes
* Idea: They are characterisic classes, used for G = U(k).
* Notation: They belong to f* Hi(Gr(n, k, $$\mathbb C$$); $$\mathbb R$$) = Hi(B; $$\mathbb R$$), and they are non-zero only if i is even, so they are often written ci(P) ∈ H2i(B; $$\mathbb R$$).

And Bundle Operations
* For the Whitney sum: c(EF) = c(E) c(F) (in terms of forms, this means exterior product).

Examples
* For SU(2):

c1(P) = 0 ,    c2(P) = −$$1\over16\pi^2$$FaFa = $$1\over8\pi$$tr(FF) .

* For a tangent bundle TM: The ci(TM) are invariants of the complex structure of M.
* For a principal fiber bundle P with curvature F: Given by ci(P) = Pi(F); In particular, if n is the dimension of the group representation,

c0(P) = 1,   c1(P) = $${\rm i}\over2\pi$$tr(F),   c2(P) = $$\big({{\rm i}\over2\pi}\big)^2$$$$1\over2$$(tr F ∧ tr F − tr FF),   cn(P) = $$\big({{\rm i}\over2\pi}\big)^n$$det F .

References > s.a. non-commutative geometry.
@ In terms of curvature, etc: Briggs gq/99.
@ And physics: Yang PT(12)jan [and quantum numbers].