Rigidity of Non-Steady Gradient Ricci Solitons

Let $(M,g)$ be a connected, compact Riemannian manifold of dimensionan n. We demonstrate that, after a suitable normalization, a shrinking gradient Ricci soliton $(M,g,f,\lambda )$ is trivial exactly when the mean value of f is less than or equal to $\frac{n}{2}$. Moreover, we prove that a normalized non-steady gradient Ricci soliton $(M,g,f,\lambda )$ is trivial if and only if its scalar curvature S satisfies the relation $S=\lambda \left(f+\frac{n}{2}\right).$ In addition, we establish that if $(M,g,f,\lambda )$ admits an isometric immersion as a hypersurface in the Euclidean space, then the soliton must necessarily be of a shrinking type. In such a case, the constant $\lambda$ and the mean curvature of M satisfy a certain inequality, with equality occurring precisely when M is isometric to a round sphere.