Infinitesimal isometries of connection metric and generalized moment map equation

Abstract

Let $(M,g)$ be a smooth Riemannian manifold, $K$ a compact Lie group and $p:P\to M$ a principal $K$-bundle over $M$ endowed with a connection $A$. Fixing a bi invariant inner product on Lie algebra $\mathfrak{k}$ of $K$, the connection $A$ and metric $g$ define a Riemannian metric $g_A$ on $P$. Let $\tilde{X}$ be the horizontal lift of vector field $X$ on $M$ and, let ${\xi }^{\nu }$ be the vertical field associated with section $\nu \in A^0(\mathrm{ad}(P))$ of the adjoint bundle. It is proved that the connection $A$ is invariant under the 1-parameter group of local diffeomorphism generated by $\tilde{X}+{\xi }^{\nu }$ if and only if $X$ and $\nu$ satisfy the generalized moment map equation ${\iota }_X{F}_A=-{\nabla }^A\nu$. The Lie algebra of fiber preserving Killing fields of $(P,g_A)$ is studied, in the case where $K$ is compact, connected and semisimple.