  We develop a general formalism for the parameter-space metric of the
multi-detector F-statistic, which is a matched-filtering detection statistic
for continuous gravitational waves. We find that there exists a whole family of
F-statistic metrics, parametrized by the (unknown) amplitude parameters of the
gravitational wave. The multi-detector metric is shown to be expressible in
terms of noise-weighted averages of single-detector contributions, which
implies that the number of templates required to cover the parameter space does
not scale with the number of detectors. Contrary to using a longer observation
time, combining detectors of similar sensitivity is therefore the
computationally cheapest way to improve the sensitivity of coherent
wide-parameter searches for continuous gravitational waves.
  We explicitly compute the F-statistic metric family for signals from isolated
spinning neutron stars, and we numerically evaluate the quality of different
metric approximations in a Monte-Carlo study. The metric predictions are tested
against the measured mismatches and we identify regimes in which the local
metric is no longer a good description of the parameter-space structure.
