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 $ξ^ν$ be the vertical field associated with section $ν\in A^0(\mathrm{ad}( P))$ of the adjoint bundle. It is proved that the connection $A$ is invariant under the 1-parameter group of local diffeomorphism generated by $\tilde{ X}+ξ^ν$ if and only if $X$ and $ν$ satisfy the generalized moment map equation $ι_XF_A=-\nabla^Aν$. The Lie algebra of fiber preserving Killing fields of $(P,g_A)$ is studied, in the case where $K$ is compact, connected and semisimple.