On Homotopic Harmonic Maps

Abstract

Let M, M′ be C ∞ Riemann manifolds such that (1.0) M is compact; (1.1) M′ is complete and its sectional curvatures are non-positive. In terms of local coordinates x = ( x 1 , … , x n ) on M and y = ( y 1 , … , y m ) on M′, let the respective Riemann elements of arc-length be and Γ i jk , Γ′ α βγ be the corresponding Christoffel symbols. When there is no danger of confusion, x (or y) will represent a point of M (or M′) or its coordinates in some local coordinate system.