On the curvature and integrability of horizontal maps (Q1081840)

Let \(\xi =(E,\pi,B,F)\) be a vector bundle over the base (differentiable) manifold B and let \(\tau_ E\) \((\tau_ B)\) be the tangent bundle of E (B). The author calls a bundle map \({\mathcal H}: \pi^*(\tau_ B)\to \tau_ E\) a horizontal map if there is a splitting of the canonical exact sequence \[ 0\quad \to \quad V_{\xi}\to^{i}\tau_ E\to^{d\pi}\tau_ B\quad \to \quad 0 \] where \(V_{\xi}\) denotes the vertical subbundle of the tangent bundle \(\tau_ E\). The goal of this article is to study the differential geometry of the horizontal map \({\mathcal H}\). The author obtains a number of necessary and sufficient conditions for a horizontal map to be integrable. Some results of \textit{P. Dombrowski} [J. Reine Angew. Math. 210, 73-88 (1962; Zbl 0105.160)] are generalized.