On gradient Riemann solitons invariant by rotation

Abstract

In this paper, we consider Riemann solitons that are conformal to an Euclidean space. Assuming that the solutions of the presented system of partial differential equations are invariant under the action of the orthogonal group, we provide all the solutions for the gradient Riemann solitons. We show that a gradient Riemann soliton (M; g) is both geodesically complete and rotationally symmetric if and only if g is the canonical product metric on R Sn????1. Furthermore, this soliton is shrinking.