Infinitesimal Isometries of Connection Metric and Generalized Moment Map Equation

The paper investigates the lifting of vector fields from a smooth manifold $$M$$ to a principal $$G$$ -bundle $$P$$ that preserves a connection $$A$$ on $$P$$ . We introduce a generalized moment map equation associated with a vector field on $$M$$ and show that the existence of a solution to this equation is both necessary and sufficient for the existence of connection-preserving lifts on $$P$$ . Using this equation, we characterize the Lie algebra of fiber-preserving Killing vector fields on the Riemannian manifold $$(P,g_{A})$$ , where $$g_{A}$$ is the Riemannian metric induced by the connection $$A$$ , a metric $$g$$ on $$M$$ and a fixed bi-invariant inner product on the Lie group $$G$$ . Finally, we solve the generalized moment map equation and present examples and applications in specific cases, illustrating the broader implications of these results.