Isometry groups of Riemannian fibrations
Let M and F be smooth finite dimensional Riemannian manifolds with isometry groups I(M) and I(F) respectively. Let G be a finite dimensional Lie group, and let P be a principal G-bundle over M with a fixed connection \(\omega\) in P. Suppose G acts on F through isometries by means of a Lie group homomorphism \(\gamma\) : \(G\to I(F)\). Then there is a canonically defined Riemannian metric on the associated fiber bundle \(E=P\times_ GF\). Let \(I_ M(E)\) denote the group of all fiber preserving isometries of E, and let \(\beta\) : \(I_ M(E)\to I(M)\) be canonical Lie group homomorphism. The main theorem of this paper identifies the kernel of \(\beta\) with a certain subgroup of the gauge group of the principal I(F)-bundle \(Q=P\times_ GI(F)\). Indeed, ker \(\beta\) is isomorphic to the isotropy group of the connection in Q associated to \(\omega\) under the action of the gauge group of Q on the set of connections on Q. This in turn is isomorphic to the centralizer in I(F) of the image of the holonomy group of \(\omega\) under \(\gamma\). This result is applied to describe \(I_ M(TM)\) where TM is the tangent bundle of M, and to characterize the Kähler condition for complex manifolds.
scientific article; zbMATH DE number 4103913