Sufficient conditions for triviality of Ricci solitons

Abstract

We found conditions on an $ n $-dimensional Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ to be trivial. First, we showed that under an appropriate upper bound on the squared length of the covariant derivative of the potential field $ \mathbf{u} $, the Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ reduces to a trivial soliton. We also showed that appropriate upper and lower bounds on the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ of a compact Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ with potential field $ \mathbf{u} $ geodesic vector field makes it a trivial soliton. We showed that if the Ricci operator $ S $ of the Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ is invariant under the potential field $ \mathbf{u} $, then $ \left(M, g, \mathbf{u}, \lambda \right) $ is trivial and the converse is also true. Finally, it was shown that if the potential field $ \mathbf{u} $ of a connected Ricci soliton $ \left(M, g, \mathbf{u}, \lambda \right) $ is a concurrent vector field, then the Ricci soliton is shrinking.

Keywords

• ricci soliton
• potential field
• geodesic vector field
• concurrent vector field