Biharmonic maps into symmetric spaces and integrable systems

TL;DR

This paper characterizes biharmonic maps from Riemannian manifolds into symmetric spaces using Maurer-Cartan forms, explicitly determining all biharmonic curves and maps from open subsets of A2 into these spaces.

Contribution

It provides a Maurer-Cartan form-based description of biharmonic maps into symmetric spaces, enabling explicit classification of such maps and curves.

Findings

All biharmonic curves into symmetric spaces are explicitly determined.

All biharmonic maps from open subsets of A2 into symmetric spaces are classified.

The biharmonic map equation is expressed in terms of Maurer-Cartan forms.

Abstract

In this paper, the description of biharmonic map equation in terms of the Maurer-Cartan form for all smooth map of a compact Riemannian manifold into a Riemannian symmetric space $(G/K,h)$ induced from the bi-invariant Riemannian metric $h$ on $G$ is obtained. By this formula, all biharmonic curves into symmetric spaces are determined, and all the biharmonic maps of an open domain of ${\Bbb R}^2$ with the standard Riemannian metric into $(G/K,h)$ are determined.