Let P be a principal O(n) bundle over a C ∞ manifold M of dimension 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 .

Let γ q,n denote the complex Stiefel bundle over the complex Grassmannian Gr n (C q ) and let ω 0 be the universal connection on this bundle. Consider the Chern character form of ω 0 defined by the formulawhere Ω 0 is the curvature form of the connection ω 0 . Let M be a manifold of dimension ≤ m and σ a closed 2k-form on M . Suppose, there exists a continuous map f 0 : M → Gr n (C q ) which pulls back the cohomology class of ch k (ω 0 ) onto the cohomology class of σ. We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map f : M → Gr n (C q ) such that f * ch k (ω 0 ) = σ.

Abstract. Starting with a short map f 0 : I → R 3 on the unit interval I, we construct random isometric map fn : I → R 3 (with respect to some fixed Riemannian metrics) for each positive integer n, such that the difference (fn − f 0 ) goes to zero in the C 0 norm. The construction of fn uses the Nash twist. We show that the distribution of n 3/2 (fn − f 0 ) converges (weakly) to a Gaussian noise measure.

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