We study the core of a proper action by a Lie group G on a smooth manifold M, extending the construction for G compact by Skjelbred and Straume (A note on the reduction principle for compact transformation groups, 1995). Moreover, we show that many properties of a proper G-action on M are determined by the action of a group \(G'\) on the corresponding core \( _cM\) . We say that such properties admit a reduction principle. In particular, we prove that a proper isometric G-action on M is polar (resp. hyperpolar) if and only if the \(G'\) -action on \( _cM\) is polar (resp. hyperpolar). In the case of a proper action by symplectomorphisms on a symplectic manifold, we show that a reduction principle holds for coisotropic and infinitesimally almost homogeneous actions. We further study the coisotropic condition for the case of a proper Hamiltonian action and its relation with the symplectic stratification described by Lermann and Bates (Pac J Math 181:201–229, 1997). In particular, we obtain several characterizations for coisotropic actions, some of which extend known results for the action of a compact group of holomorphic automorphisms on a compact Kähler manifold obtained by Huckleberry and Wurzbacher (Math Ann 286:261–280, 1990). Finally, we study some applications of the core construction for the action of a compact Lie group on a Kähler manifold by holomorphic isometries.
