Study of harmonic and biharmonic maps on Thurston geometry

Abstract

This thesis deals with the study of Harmonic and Biharmonic maps on Thurston geometry. The aim of this thesis is to classify harmonic and biharmonic applications in Thurston model geometries of dimension 3. Three-dimensional Thurston model geometries are classifiedby W. Thurston, this classification has eight Three-dimensional model geometries, to know, E3, S3, H3, S2 × R, H2 × R, Sl ^2(R), Nil3 and Sol3. Also we classify harmonic and biharmonic applications in Thurston model geometries of dimension 4. R. Filipkiewicz classified the Thurston geometry of dimension four and he proved that there are 19 classes of maximal geometries in 4-dimension, to know, E4, S4, H4, P2(C),H2(C), S2 × S2, S2 × E2, S2 × H2, H2 × E2, H2 × H2, H3 × E1, H3 × E1, Nil4, Solm;n 4 , Sol04, Sol4 1, F4, Sl ^2(R) × E1 and Nil3 × E1. In dimension 3 we study biharmonic Legendre curves on three-dimensional Lorentzian Heisenberg space (H3; g) and we study biharmonic curves in three-dimensional generalized symmetric spaces. We also show that a linear map from an Euclidean space in three-dimensional generalized symmetric spaces is biharmonic, and we give a complete classification of such maps. In dimension 4 we study harmonic and biharmonic applications in Thurston geometry of dimension 4. We introduce the 4-dimensional geometry Nil4 and we define the metric gNil4. We give the Christoffel symbols and the Riemannian curvature to study the biharmonic curves in Nil4 space.