Biharmonic maps into Sol and Nil spaces

TL;DR

This paper investigates biharmonic maps into Sol and Nil spaces, characterizing non-geodesic biharmonic curves, proving the non-existence of certain biharmonic helices, and classifying linear biharmonic maps from Euclidean space.

Contribution

It provides a complete classification of linear biharmonic maps into Sol and Nil spaces and characterizes biharmonic curves in these geometries.

Findings

No non-geodesic biharmonic helix exists in Sol space.

Linear maps into Sol or Nil are biharmonic iff they are harmonic.

Complete classification of linear biharmonic maps from Euclidean space.

Abstract

In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Euclidean space into Sol or Nil space is biharmonic if and only if it is a harmonic map, and give a complete classification of such maps.