On the minimality of the corresponding submanifolds to four-dimensional solvsolitons

Abstract

In our previous study, the author and Tamaru proved that a left-invariant Riemannian metric on a three-dimensional simply-connected solvable Lie group is a solvsoliton if and only if the corresponding submanifold is minimal. In this paper, we study the minimality of the corresponding submanifolds to solvsolitons on fourdimensional cases. In four-dimensional nilpotent cases, we prove that a left-invariant Riemannian metric is a nilsoliton if and only if the corresponding submanifold is minimal. On the other hand, there exists a four-dimensional simply-connected solvable Lie group so that the above correspondence dose not hold. More precisely, there exists a solvsoliton whose corresponding submanifold is not minimal, and a left-invariant Riemannian metric which is not solvsoliton and whose corresponding submanifold is minimal.