We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all sa
tisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.
We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanv{z}urova are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.
We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant
of one half the distance function from a fixed point.
