  Harmonic maps from S^2 to S^2 are all weakly conformal, and so are
represented by rational maps. This paper presents a study of the L^2 metric
gamma on M_n, the space of degree n harmonic maps S^2 -> S^2, or equivalently,
the space of rational maps of degree n. It is proved that gamma is Kaehler with
respect to a certain natural complex structure on M_n. The case n=1 is
considered in detail: explicit formulae for gamma and its holomorphic
sectional, Ricci and scalar curvatures are obtained, it is shown that the space
has finite volume and diameter and codimension 2 boundary at infinity, and a
certain class of Hamiltonian flows on M_1 is analyzed. It is proved that
\tilde{M}_n, the space of absolute degree n (an odd positive integer) harmonic
maps RP^2 -> RP^2, is a totally geodesic Lagrangian submanifold of M_n, and
that for all n>1, \tilde{M}_n is geodesically incomplete. Possible
generalizations and the relevance of these results to theoretical physics are
briefly discussed.
