> as lemmas

1934 Ann. Math. 35 875 We shall deal exclusively with..metrized, topologically complete vector spaces in which distance is invariant under simple translations.

1990 Jrnl. Amer. Math. Soc. 3 723 The condition for the metric to be harmonic is G_K = 0. If this condition holds..we get a metrized Higgs bundle.