Connections with prescribed first Pontrjagin form

Abstract

Let P P be a principal O ( n ) O(n) bundle over a C ∞ C^\infty manifold M M of dimension m m . If n ≥ 5 m + 4 + 4 ( m + 1 4 ) n\geq 5m+4+4\binom {m+1}{4} , then we prove that every differential 4-form representing the first Pontrjagin class of P P is the Pontrjagin form of some connection on P P .