Construction of connection inducing maps between principal bundles. I
Giuseppina D’Ambra · Journal of Differential Geometry · 1987
Consider two C°°-sinooth principal bundles, say X -> V and Y -> W, with the same structure group G and with C°° connections Γ on X and Δ on Y, respectively.We look for a C°°-map /: V -> W such that the induced bundle f*{Y) over V with the induced connection /*(Δ) is isomorphic to (X, Γ).This means that / can be covered by (or lifted to) a morphism of bundles, F: X -> Y, inducing Γ from Δ, which is expressed by F*(Δ) = Γ.0.1.The problem of inducing connections was first studied by Narasimhan and Ramanan [3] who proved that for a given compact Lie group G and an integer n = 0,1, , there exists a (universal) bundle (Y, Δ) over some (classifying) compact manifold W, such that every G-bundle X over an ndimensional manifold V with an arbitrary C 00 -connection Γ can be induced by a C 00 -morphism F: X -> Y. Furthermore, they give a precise description of the universal connection Δ for the unitary and the orthogonal groups.Namely, if G = U(p) they take the Grassmann manifold Gτ p (C q ) for W and use the standard connection Δ on the canonical bundle Y -> Gτ p (C q ) (here Y is the Stiefel manifold of orthogonal /?-frames in C q ).The dimension q for which they prove the existence of F is q = (n + 1)(2« + I)/?3 , where n = dimF.Similarly, for G = O(p), their method provides a connection inducing map into the real Grassmann manifold Gτ p (R q ), again for q = (n + l)(2n + I)/? 3 .0.2.The result by Narasimhan-Ramanan was improved for G = O(p) by Gromov (see 2.2.6 in [1]) who showed the existence of a connection inducing map /: V -> Gτ p (R q ) for q = max(/?(«+ 1), p(n + 2) + n).Furthermore, if the manifold V is parallelizable and the bundle X -> V is trivial, then