TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 9, Pages 3813–3824 S 0002-9947(03)03311-7 Article electronically published on May 15, 2003 CONNECTIONS WITH PRESCRIBED FIRST PONTRJAGIN FORM MAHUYA DATTA Abstract. Let P be a principal O(n) bundle over a C ∞ manifold M of di- mension m. If n ≥ 5m +4+4 ( m+1 4 ) , then we prove that every differential 4-form representing the first Pontrjagin class of P is the Pontrjagin form of some connection on P . 1. Introduction Let P be a principal O(n) bundle over a C ∞ manifold M of dimension m, and let p i ∈ H 4i (M ) denote the i-dimensional Pontrjagin class of P . We address the question whether a 4i-form representing the class p i is a Pontrjagin form of some connection on P . In [1] we considered the top-dimensional Pontrjagin class p d of a principal O(n) bundle P over a 4d-dimensional open manifold M for n ≥ 2d, and we gave a homotopy classification of connections α on P that satisfy p d (α)= ω, where ω is a volume form on M . In this paper, we take up the case of the first Pontrjagin form and prove the following result. Theorem 1.1. If n ≥ 5m +4+4 ( m+1 4 ) , then every differential 4-form represent- ing the first Pontrjagin class p 1 is the Pontrjagin form of some connection on P . Moreover, when M is a closed manifold, the same is true for n ≥ 5m +4 ( m 4 ) . Here ( m k ) denotes the integer m! k!(m−k)! . We observe that when n>m, then P reduces to the direct sum P 1 ⊕ P 2 of two principal bundles, where P 1 is an O(m) bundle and P 2 is the trivial O(n − m) bundle on M . Since the Pontrjagin form is additive, the above observation reduces the problem to finding a connection on a trivial principal bundle with a given exact form as its Pontrjagin form. Now, if an exact 4-form on M can be expressed as the sum of q primary mono- mials of the form df 1 ∧ df 2 ∧ df 3 ∧ df 4 , where the f i ’s are smooth functions on M , then we can explicitly construct a connection on the trivial principal O(2q)-bundle over M by taking a 2 × 2 block α =  0 f 1 df 2 − f 3 df 4 −f 1 df 2 + f 3 df 4 0  along the principal diagonal for each such monomial. It can be seen easily that the Pontrjagin form of such a connection is the given exact form on M . Indeed, we can prove the following result (compare ([2], 3.4.1 (B ′ ))). Received by the editors September 26, 2002 and, in revised form, February 14, 2003. 2000 Mathematics Subject Classification. Primary 53C05, 53C23, 58J99. Key words and phrases. Principal bundle, connections, first Pontrjagin form. c 2003 American Mathematical Society 3813 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use