Splitting theorem for Ricci soliton

Abstract:

Let $(M, g, f)$ be a gradient Ricci soliton $\nabla ^2 f+Ric=\lambda g$ with $\lambda \in \{\frac {1}{2}, 0, -\frac {1}{2}\}$. Suppose there is a geodesic line $\gamma : (-\infty , \infty )\rightarrow M$ satisfying \begin{eqnarray*} \liminf _{t\rightarrow \infty }\int _0^{t}Ric(\gamma ’(s), \gamma ’(s))ds +\liminf _{t\rightarrow -\infty }\int _{t}^{0}Ric(\gamma ’(s), \gamma ’(s))ds \geq 0, \end{eqnarray*} then $(M, g, f)$ splits off a line isometrically.