  We study non-linear sigma models on target manifolds with constant (positive
or negative) curvature using the functional renormalization group and the
background field method. We pay particular attention to the splitting Ward
identities associated to the invariance under reparametrization of the
background field. Implementing these Ward identities imposes to use the
curvature as a formal expansion parameter, which allows us to close the flow
equation of the (scale-dependent) effective action consistently to first order
in the curvature. We shed new light on previous work using the background field
method.
