Abstract
We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M, g, w, sigma) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r(2) <= n sigma r on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue lambda satisfying n(2)lambda(2) >= r(2); the third result uses a suitable lower bound on the Ricci curvature S(w, w). Finally, we show that an n-dimensional connected Ricci almost soliton (M, g, w, sigma) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, n sigma - r is a constant along integral curves of w and the Ricci curvature S(w, w) has a suitable lower bound.