Riemannian moment map (Q1006903)

In [Philos.~Trans.~R.~Soc.~Lond., A 308, 523--615 (1983; Zbl 0509.14014)], \textit{M.~Atiyah} and \textit{R.~Bott} discovered that the curvature as a function on the space of connections on a principal \(G\)-bundle over a Riemann surface can be interpreted as the momentum for the gauge group action on the space of connections. This observation, together with its several extensions, has proved to be an extremely useful framework to study various kinds of geometries. The goal of this paper is to find such a framework in the Riemannian world which extends its counterpart in the Kählerian world. The author introduces a notion of a moment map for a real reductive group acting on a Riemannian manifold, and sets up the foundation for the theory of Riemannian moment maps. This extends the well-known theory in Kähler geometry. Both finite and infinite dimensional examples are presented.