Given a regular scheme $X$, projective and flat over $Spec(\mathbb{Z})$, i.e. an arithmetic variety and a vector bundle $E$ on $X$. We get an associated vector bundle $F$ on the associated complex manifold $M:=X(\mathbb{C})$. On $M$ one has complex conjugation $c$.

Following Kobayashi's "Differential geometry of complex vector bundles", the set $H(F)$ of hermitian metrics on $F$ can be identified with the symmetric space $GL(F)/U(F)$.

How to find those metrics $h \in H(F)$, which are invariant under $c$, i.e. $c^{*}h=h$? Is there a nice description of this set?

I am interested in this metrics, because they are often mentioned in articles about Arakelov geometry, especially Soule's book. This condition comes somehow out of nowhere. I would be very happy to understand why this metrics are the right ones to consider. Another question in this context: is "invariant under $c$" a strong restriction to the metrics in $H(F)$, i.e. are there many of them or rather a few who fulfill this condition.