Topology and its Applications
An action of a Lie group G on a bundle is said to be transverse if it is projectable and if the orbits of G on E are diffeomorphic under π to the orbits of G on M. Transverse group actions on bundles are completely classified in terms of the pullback bundle construction for G-invariant maps. This classification result is used to give a full characterization of the G invariant sections of E for projectable group actions.
Ian M. Anderson, Mark E. Fels, Transverse group actions on bundles, Topology and its Applications, Volume 123, Issue 3, 30 September 2002, Pages 443-459, ISSN 0166-8641, 10.1016/S0166-8641(01)00211-5.
Published by Elsevier in Topology and Its Applications. Author preprint available for download through link above.