  This note is a study of nonnegativity conditions on curvature which are
preserved by the Ricci flow. We focus on specific kinds of curvature conditions
which we call noncoercive, these are the conditions for which nonnegative
curvature and vanishing scalar curvature doesn't imply flatness. We show that,
in dimensions greater than 4, if a Ricci flow invariant condition is weaker
than "Einstein with nonnegative scalar curvature", then this condition has to
be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow
invariant curvature condition which is stronger than "nonnegative scalar
curvature" cannot be (strictly) satisfied by compact Einstein symmetric spaces
such as S^2xS^2 or CP^2. We also investigate conditions which are satisfied by
all conformally flat manifolds with nonnegative scalar curvature.
