On the Triviality of a Certain Kind of Shrinking Solitons

Abstract

In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue has multiplicity at least $n−1,$ the triviality are also true if we naturally require the positivity of Ricci tensor.
We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded from above, then the soliton are Einstein.