Chern-Weil theory
/ churn-VYE (Chern as in 'churn'; Weil as 'vye') /
Chern-Weil theory is the alchemy that turns a connection's curvature into topology. Pick any bundle, choose any connection on it, compute its curvature, feed the curvature into a suitable polynomial — and out comes a closed differential form whose cohomology class does not depend on the connection you chose. The curvature is geometry (it changes if you change the connection), but a special recipe extracts from it something that is pure topology of the bundle. That something is the characteristic classes.
Precisely: let G be the structure group with Lie algebra g, and let P be an invariant polynomial on g — a polynomial function on g that is unchanged under the adjoint action of G (for matrix groups, think of symmetric functions of eigenvalues, like trace or determinant). Given a connection with curvature 2-form Omega (a g-valued 2-form), substitute Omega into P to get an ordinary differential form P(Omega) on the base. Two facts make the magic: (1) P(Omega) is closed, d P(Omega) = 0, which follows from invariance of P plus the Bianchi identity; and (2) its de Rham class [P(Omega)] in H^*(M; R) is independent of the connection — interpolate between two connections by a line A_t = A_0 + t(A_1 - A_0) and show the difference is exact (a transgression form). So the map sending an invariant polynomial to a cohomology class, the Chern-Weil homomorphism, is well defined on the bundle alone.
This is the differential-geometric construction of characteristic classes: applying it to the right invariant polynomials of the curvature of a complex bundle gives the Chern classes, of a real bundle the Pontryagin classes, and the Pfaffian gives the Euler class. Its great virtue is computability — you get explicit closed forms you can integrate, which is how Gauss-Bonnet and index theorems acquire local curvature formulas. Two honest limitations: first, Chern-Weil produces classes in real (de Rham) cohomology only, so it is blind to torsion that the topological definitions over the integers detect; a Chern-Weil form can be exact while the integral class is nonzero torsion. Second, it requires a smooth bundle and connection — for the purely topological theory over Grassmannians and K-theory you need the homotopy-theoretic definition, which lives in another field.
For a complex vector bundle with curvature Omega, the invariant polynomial det(I + (i/2pi) Omega) expands as a sum of forms of degrees 0, 2, 4, ...; the degree-2j piece is a closed 2j-form representing the j-th Chern class c_j. The simplest case, a complex line bundle, gives the single 2-form (i/2pi) Omega representing c_1 — and integrating it over a closed surface returns an integer, the degree of the bundle. Different connections give different 2-forms but always the same integer.
det(I + (i/2pi)Omega) packages all the Chern classes at once; its pieces are closed forms whose classes ignore the connection.
Chern-Weil computes characteristic classes in real (de Rham) cohomology only — it is blind to torsion. A class that is pure torsion in integral cohomology shows up as zero in Chern-Weil, so a vanishing Chern-Weil form does not prove the integral class vanishes. For the full integral/torsion story you need the topological (Grassmannian/K-theory) definition.