  In this paper, we prove some rigidity results for both shrinking and
expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons
are Einstein if we control the maximum value of the potential function. Then,
we prove some rigidity results for non-compact gradient expanding and shrinking
Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic
condition on the scalar curvature at infinity.
