Euler and Pontrjagin currents of a section of a compactified real bundle

AbstractThe canonical family of Thom forms τs for 0 < s < ∞ constructed by Harvey and Lawson on an oriented real vector bundle V → X with metric connection Dv is shown to have an extension to the bundle of real projective spaces, P(R ⊕ V) → X, which compactifies V in the fibre directions. The current limit as r → ∞ and s → 0 of the smooth transgression formula τr − τs = dσr,s is the current equation x̃(Dv) + [P(V)] Res − [X] = dσ∞,0 on P(R ⊕ V). Here σ∞,0 is an Lloc1 current on P(R ⊕ V) and [X] is the current of integration over the zero section of V. If the rank of V is even, x̃(Dv) is the twisted extension to P(R ⊕ V) of the Chern-Euler form of the connection Dv on V, [P(V)] integrates densities over the nonorientable submanifold P(V) of P(R ⊕ V) and Res is a smooth twisted form on P(V). If the rank of V is odd, x̃(Dv) ≡ 0, [P(V)] is the current of integration over P(V) and Res is a smooth form on P(V).This construction can be pulled back to X by an atomic section ν:X → P(R ⊕ V) satisfying an appropriate orientability hypothesis to obtain a current equation on X relating the zero and pole sets of ν to the topology of the bundle. Applications to the differential topology of maps into spheres and real projective spaces are discussed