Geometric applications of harmonic maps

Abstract

In this section, we shall use our Bochner type formula (3.2.10),

$$[tex]{\Delta ^ - }e\left( f \right) = |\nabla df{|^2} + \frac{1}{2} < df \cdot Ri{c^N}\left( {{e_\alpha }} \right),df \cdot {e_\alpha } > - \frac{1}{2} < {R^M}\left( {df \cdot {e_\alpha },df \cdot {e_\beta }} \right)df \cdot {e_\beta },df \cdot {e_\alpha} > [/tex]$$

((5.1.1))

for a harmonic f: N → M in order to derive some elementary results about the topology of nonpositively curved Riemannian manifolds. These results are well-known, and the present section therefore is included only for reasons of exposition.