Functional renormalization of N scalars with O(N) invariance

We discuss general theories of N scalar fields with O(N) symmetry. In addition to the standard case of linearly realized symmetry there are also examples that carry nonlinear realizations, with the topology of a cylinder R×SN−1 or a sphere SN. We write flow equations for the theory in the second order of the derivative expansion in the background field and discuss the properties of scaling solutions with vanishing potential.