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 $R\times S^{N-1}$ or a sphere $S^N$. 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.