AFRICA MATHEMATICS ANNALS

On Jacobi fields along harmonic maps with values into a symmetric Riemannian manifold

Moussa KOUROUMA
UFR Mathématiques - Informatique
Université de Cocody, 22 BP 582 Abidjan 22, Côte d'Ivoire

Mathematics Subject Classification: [2010] 58C, 58E, 49R50, 35J, 35D .
Key words: Riemannian manifold, geodesic, harmonic map, Jacobi field, symmetry.

Abstract:

We prove essentially that : for a Riemannian manifold (M,g) and a symmetric Riemannian manifold (N,h), if[0,1]×M ∋ (t,x)→ u_t(x) ∈ N is in W^1,2(M,N) and is such that, for any t ∈ [0,1], u_t is a harmonic map from (M,g) into (N,h), which is smooth on a subset U ⊆ M such that vol(M\U) = 0, and is such that t→ u_t(x) is constant for x ∈ ∂M, then the normed velocity vectorfield is parallel, i.e.
∇[||[(∂u_t)/(∂t)]||⁻¹[(∂u_t)/(∂t)]](t,x) ≡ 0.