On Homotopic Harmonic Maps

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 = (x1, … , xn) on M and y = (y1, … , ym) on M′, let the respective Riemann elements of arc-length be and Γijk, Γ′αβγ 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.